First off, I'm already resigned to the fact that I am a failure at soldering. (hours of research into dozens of website/youtube how-tos, 4 different soldering irons (albeit inexpensive), 9 or 10 different tips, fluctuating flux amounts, 3 different brands of solder, innumerable attempted techniques, 2 or 3 permanent tiny scars from sub-dermal blistering, and a small pile of charred electronics had, some months ago, broken my will and led to my surrender). But I'm still left with an unproven theory that others here might help confirm or reject. Being that, when faced with being placed onto a targeted location, the solder--after assuming a spherical shape--would, at times, refuse to separate from the iron's tip or, more often, flee the area, rolling the length of the workspace, around obstacles, up walls, or--I imagine--down entire city blocks until I would decide to break off the chase and simply create a whole new obstinate silvery blob, could it be that I had inadvertently magnetized the soldering tip by storing it next to a battery or some other atmospheric anomaly and thereby causing it to disrupt the normal soldering process and ultimately revealing it to be the cause of my circuitrous disability? While I eagerly await your replies I will click on the "unwanted magnetic induction" question to the right that I just now discovered. My gratitude to the team, --smf
Adding to all previous comments, my experience is to avoid cheap (hobbyist) irons altogether as they do not reach a high enough temperature and cannot maintain that temperature when the tip is presented to the job very well. A tip that is clean and tinned (but not running with solder) should be presented to heat the joint slightly prior to the solder being applied and if the iron is hot enough and the component and board are clean, the solder will flow and it should be possible to do it in around a second. There should be no balling at all. My advice is to invest in a Weller or similar make iron if you are doing a lot of soldering and avoid the cheap ones altogether.
If you have a clean iron tip but (very) dirty target surface, the solder will "stick" to the iron... well within reason. If you add a lot solder you'll get a large blob rolling. It seems this is the scenario you're describing, although sans a short clip (which you could upload to youtube), it's hard to tell exactly what's going on there. But if you follow the advice below and clean both the iron and target surface (or buy those that don't need extraordinary cleaning measures), you should be fine.
A cheap (not temperature controlled) tip is usually covered in copper. Solder creates an alloy at the surface when you "tin you tip", this is why it sticks to it. If you add a large amount of solder to the tip (not a recommended technique by the way), the blob that forms is kept in place by surface tension because the molten solder acts as a liquid; there aren't any significant magnetic forces involved. If your iron tip gets too hot (or even if it's not too hot but let it on for a long time) you get an oxide layer over it; this oxide becomes visibly black after it gets thick enough. The oxide layer prevents (additional) solder from adhering to the iron, so you get those spherical blobs because of surface tension. You need to clean an re-tin the tip periodically.
The same issue, oxidation, can affect the targets surface, so when that happens you get to chase blobs there too. You're not saying exactly what you're soldering on. Assuming it's a cheap copper protoboard that was not covered in anything protective, you need to clean it thoroughly with (separate) flux solution and possibly even scraping it, although the latter can result in uneven copper. There are some reviews of ad-hoc board rubbers... and the damage they cause... and the residues they leave behind. For single-pad protoboards, it might be ok, but when using a protoboard with any kind of pre-made strips, I stay away from rubbing/scraping. Furthermore you cannot rely on abrasion alone (i.e. just scrubbing/rubbing) to clean copper well enough for soldering; you still need flux (preferably in your solder wire) to deoxidize at microscopic level and keep the copper oxide-free right until the alloy forms.
The best solution (in terms of time wasted) is not to buy the really cheap protoboards. The semi-decent ones are covered in organic solderability preservative (OSP) which prevents the copper from oxidizing (majorly) until you get to work (example). I can solder directly to FR2 OSP'd boards (sans any cleaning) even after a couple of years in storage. The (2% or so) flux contained in the typical hobbyist flux-core solder wire is enough cleaning for those.
The more deluxe version of protoboards (typically those made of FR4) have all the pads pre-tinned (usually through hot-air leveling) and have solder mask between pads; (example). These boards are basically instant success at soldering and have long storage life (they're like post-production PCBs). As this comparative presentation of OSP and HASL says (regarding the advantages of HASL/pre-tinned boards): “Nothing Solders Like Solder”, which reflects my own experiences. However, I personally find the added cost of FR4 HASL'd protoboards is generally only worthwhile if you have at least some surface-mount components involved, and then I prefer to buy combo THT/SMD boards (example) [for hand soldering] because cause I'm rather fat-fingered when it comes to tiny surface-mount passives (resistors etc.).
Proper soldering technique matters too, of course; see https://youtu.be/AY5M-lGxvzo?t=102 for instance. But you've probably seen something like that already. The magic that you see in that clip involves a pre-tinned board. A typical example of amateur soldering on (non-tinned) copper board is https://www.youtube.com/watch?v=jr2Z_fztpxg&t=300 You can see it takes longer to make each joint and sometimes the solder doesn't cover the whole pad immediately. The essence of the technique [if you still need a repetition of that] is to have both things you want to solder (e.g. pad and lead) heated up (with the iron's tip) and then touch the joint with the solder wire. Better/faster heat transfer is enabled by a small amount of liquid solder on the iron tip (increasing the contact surface), but this shouldn't be your main source of solder for making the joint.
Let me say finally say that an expensive iron alone (even temp-controlled) will not be a substitute for a clean target surface. It would be a foolish way to allocate your budget if you keep using cheap/oxidized boards. Furthermore, even on a clean surface, temp-control of the tip is nowhere near the ultimate factor in success/failure of the joint; you need consider the thermal mass the whole assembly. Then again, you haven't told us what you're soldering.
And yeah, you can get good results with a totally cheap iron, prolly under $4 by the looks of it... on pre-tinned boards and with a modicum of technique. That guy reminds me how built by first radio kit as a teenager.
No, magnetism very unlikely has anything to do with your soldering problems. Solder is not magnetic, and storing a soldering iron near a battery is meaningless, as batteries aren't magnetic either.
You can run into magnetism issues with soldering guns, which are quite different from soldering irons. With a gun, the "tip" is heated by directly running current thru it. The current has to make a about-face at or near the tip, which causes a meaningful magnetic field since the current is so high, even though it's just a 1 turn. Even then, it has little effect on solder or copper. The only issue I've had with a gun related to magnetism is that some transistor leads are steel inside, and they would be attracted to the bend of the current.
That all said, it seems your real problem is lack of practice. Nobody is born being able to solder well. No matter how much you read or watch videos, there is no substitute for doing it, and doing a lot of it so that you develop your own intuition and "feel" for how solder behaves and flows.
You talk about large balls of solder stuck to the iron. That implies you are feeding the solder on the iron instead of the heated part. That also means the part itself was too cool, not clean enough, or no flux got onto the part.
It's tempting to feed solder onto the iron because it melts nicely, but that won't necessarily make it flow onto the part. The right technique is to hold the iron firmly against the part, then feed just a little solder as best you can between the iron and the part. That solder isn't meant to do any soldering, but to reduce the thermal resistance between the iron and the part. This also applies a little flux in there, which should allow the solder to flow between the iron and the part. Once that happens, the part will get heated by the iron. Wait a second or so, then feed solder into the joint that you are trying to connect. A little of the first solder should already have wicked there, so the additional solder should now flow nicely around everything and "fill" the joint. Remove the iron while keeping everything still and blow on the joint gently until it hardens. You can see that happening by the solder turning from a nice and shiny silver look to a dull silver look. This shouldn't take more than a couple of seconds.
A temperature controlled iron is a really useful tool for soldering. A fixed wattage or even "adjustable" iron may seem like a good entry level tool, but these are much more difficult to use well. Last time I looked around, the Weller WES51 was the cheapest temperature controlled soldering iron out there. It can usually be found at about $100 street price new. This is worth it even for medium level hobby soldering.
It is clearly better to use the best tools available.
I have had success using a sub 5 GBP soldering iron, but I knew how to solder already. Very good soldering irons are the best way to go, however, IMHO it is a bit like bicycle stabilisers, they give extra confidence and mitigate against some types of accident, but once the technique is good, it is less important (unless you plan on doing a lot of soldering).
If you can track down access to a good soldering iron (maybe you are at college), then practice using that. You won't necessarily need anything as expensive once you've 'cracked it.
Better yet, try to practice with someone who knows how to solder. They will likely see some of the errors and quickly help you to fix them.
As everyone has explained, you problems sound like 'dirt', and not magnetism. Cleanliness is key.
I strongly recommend you start with a flux-core tin/lead solder. Tin/Lead solder is much easier to use than lead-free solder. I've taught many people, including children (under 11) to solder, and the difference between 60/40 tin/lead and lead-free solder is extremely noticeable; beginners find tin/lead solder is much easier. Always wash your hands after you handle lead, and don't use lead solder paste.
Clean the thing you are soldering. Don't use water. Use an 'abrasive rubber block', fine emery paper, or clean wire wool (not the type with soap in).
A flux pen can help too, though my experience is cleaning immediately before soldering works well.
I'd recommend a modest-price temperature controlled iron, for example, lots of suppliers, even shops, seem to have this soldering station under their own brand. They are so generic, they often pop up in shop's sales for under 40GBP.
Once you get the technique then you can move on to lead-free solder with silver (which is a bit harder for beginners to use than leaded solder), then once you have that, move onto lead-free. I spend, maybe an hour with people, and the vast majority have the technique within the hour. Some people are a bit diffident, and spend too much time heating the joint, or not enough. It takes some practice to get the 'rhythm', iron-in heat, solder-in melt, solder-out, iron-out.
The most common errors are:
Trying to carry solder to the joint on the soldering iron tip, so the joint isn't heated, and the solder flux has boiled off, producing balls of solder which won't coat the joint surfaces.
Heating the wire, but not the copper track, so the solder 'globs' around the wire only. Dirty surfaces, so the solder 'globs' and doesn't 'wet' the surfaces.
Heat applied to the track for too long, so that the track peels off.
People have given good advice on successful soldering.
Prepare to do lots of practice. I use stripboard (cheap knock-off veroboard) and ordinary through-hole resistors. Both are relatively cheap, so many hundreds of joints are only a couple of GBP.
Get a magnifying glass, or 'watch makers eye glass' (a few GBP) and inspect each joint every couple of joints. Look to see if the solder has melted and coated the two surfaces. With tin/lead solder it should look shinny, unleaded solder has a duller, less obviously good, finish.
Best of luck.