Some caps -- such as nearly all electrolytic capacitors and tantalum capacitors -- are polarized. Such caps use some sort of chemical reaction between an anode and a cathode made of two different kinds of materials to form a thin insulating layer. When you hold one of these caps in your hands, you will see a "-" mark by the pin intended to stay more negative, or a "+" mark by the pin intended to stay more positive.
If a polarized cap is ever "reverse biased" more than 1 V to 1.5 V (typical), it drives that chemical reaction in reverse, eating away at the thin insulating layer, leading to a short between the two pins.
Not only is that capacitor no longer working,
after that, any significant voltage -- forward or reverse -- could make that "capacitor" overheat and in some cases explode.
The person drawing the circuit and connecting the capacitor in a circuit must make sure the "+" end goes towards the more positive voltage, and the "-" end goes towards the more negative voltage, at all times, to prevent catastrophe.
See the Wikipedia article Greg pointed out for more details.
Other caps -- such as nearly all ceramic capacitors, paper disk capacitors, and mica capacitors -- are non-polarized.
Such caps typically use an anode and a cathode made of identical metal, and they work just as well with "reverse biased" voltage as forward biased.
They don't have either "+" or "-" mark, because they don't need one.
& 3. You never "need" a polarized cap. Practically all physical circuits would work just as well, and perhaps better, if the polarized caps were all replaced with non-polarized caps of the same capacitance and voltage rating.
The opposite is not true -- you often can't replace non-polarized caps with polarized caps.
Some circuits require a capacitor that can handle a high positive voltage at some times and a high negative voltage at other times (polarity reversal), which requires a non-polarized capacitor.
The only reason people use polarized caps is because they often cost much less than non-polarized caps of the same capacitance and voltage rating.
However, when drawing a schematic, you should always draw a "+" sign on one side of a cap whenever you intend that that the cap always has positive voltage applied to it, it never suffers polarity reversal.
That helps the people reading the schematic understand what you meant.
That gives people putting together the physical circuit the option of using polarized capacitors, even though many times it is more convenient to use non-polarized capacitors in the place of the polarized capacitors clearly marked on the schematic.
It also tells people putting together the physical circuit, should they choose to use a polarized capacitor, which way around the polarized capacitors should go.
It also communicates to repair people that, if they measure a negative bias voltage, that something has gone horribly wrong.
The schematic you show -- with the clearly marked "+" polarized capacitor -- would work just as well with a non-polarized capacitor.
The "+" on one end of the capacitor is telling us that that end is expected to never be negative relative to the other end.
It's also telling us that we have the option of using a polarized or nonpolarized cap in that location when we build that circuit.