Why do home appliances like a light, TV, and fan use AC power directly from the supply, but a phone and laptop use an adapter to convert AC to DC?
This is mostly due to the line voltage being "high" compared to what normal electronics needs, and what is safe.
Devices that use significant power, like lights and fans, are built to run directly off the power line voltage. The cost of protecting the user from the higher voltage is offset by the more efficient use of the power and not having to convert it.
Devices with electronics in them need low voltage to run the electronics. This means a extra power supply, which converts the high line voltage to the low voltage needed by the electronics. It also converts to DC, because the electronics need DC to operate.
Given that there is this converter between the line voltage and the actual internals of the device, manufacturers have a choice about where to put it. In some cases, it makes sense to put this converter external to the device. That alleviates the need to protect the user from high voltage at the device. I go into more detail here.
Some electronic devices are large enough that it makes sense to put the power supply inside. Your TV example is in this category. It's still a electronic device that internally runs on low voltage DC, but it's big enough that the power supply is internal.
You have to start from "why is the voltage of power network is 110VAC/220VAC". So it's AC because this way it's easy to convert voltage by transformers or to move motors (AC). Both reasons were there long before electronics came around.
For electronics, like TV, laptop, whatever, you will always need DC, and significantly lower than 160V/315V (rectified AC). This is because how semiconductors work- unlike vacuum tubes (which i have no idea about, i am not so old).
So why do we have sometime power supply inside a device and sometimes outside? It's because application considerations. Laptop- you don't want to have a power supply inside when you are on the move. So you use something external, that you can leave at home.
For TV it doesn't matter, so normally it's better for the customer to have it inside. Although sometimes for extremely slim devices the power supply will still be external.
I have a feeling that even though most of the explanations given here are mostly correct, they're not really answering what the OP asked. Which is funny, because I think the explanation should only be a few sentences long.
The main reason AC comes out of your outlet, is because it is very inefficiënt to transport DC over long distances (and especially low voltage DC). (Besides that: the electricity is also AC when it is generated, for most sources at least.) When it comes in your home, it is pretty much always converted to DC before use, because almost all modern devices need DC. The ones that need some kind of AC will generate their own, and not use the AC supplied to it from the outlet.
There are only a few exceptions. (I can think of only 3 for the moment, but there's probably more):
motors: most (large) elektromotors run on AC, so they can use it straight away from the outlet.
incandescant lamps: they require heat to make the lamp glow. Wheter that heat comes from AC or DC doesn't matter, so there's no reason to convert it.
heating elements: same as for incandescant lamps
Your TV is not one of the exceptions, because it converts to DC internally, same as your laptop does with an external adapter.
Another reason for having a separate AC to DC converter is Safety. Devices like laptops and mobiles are typically meant to be handled by the user with close contact to the skin. A small fault with an internal AC to DC converter can be very dangerous.
The easiest solution is to have the line voltage handling outside and far away from the device with proper isolation in the adapter. The device only sees the low voltage DC.
The home is supplied with AC because regular pole transformers and their bigger cousins require AC, and most early home electrical devices didn't care whether they got AC or DC. And for a number of other reasons it's easier to deal with AC in the distribution process.
Pretty much all devices that process audio or video (including old tube-type radios and TVs) use DC internally for most of their operations. But converting from AC to DC on a small scale is not particularly difficult.
Part of the reason is historical. Voltage conversion used to involve much larger and more expensive hardware than it does today, and many electrical gadgets used to require higher voltage than they do today. White LEDs, which are the first general-purpose electric lamp design that doesn't need a high voltage (either for efficiency or to make it work at all), only came on the market circa 2010. Linear "wall warts" were still commonplace in 2000, and so were cathode-ray-tube TVs. Radios and TVs that used vacuum tubes internally (perhaps alongside transistors) were still commonplace in the early 1980s.
Therefore, older uses of electricity, such as lamps and household appliances, tend to have been designed to run directly on the line voltage, both because availability of line voltage was more useful to them, and because it was cheaper to do it that way.
Any household device that you purchase today, that still runs directly on the line voltage, probably has a high-power motor or heater inside; those are the applications for which it's still more efficient to use line voltage directly. LED light bulbs are a special case: it would make a certain amount of sense to feed them 12VDC or so, but because houses already have all the wiring and lamp hardware set up for line-voltage lighting, they have embedded power converters instead.
(Question for the peanut gallery: What voltages are used internally inside current-generation consumer-grade audio gear? Those do also tend to take a line-voltage power input, but that might just be the "we can fit the converter inside the case" scenario mentioned in other answers. I don't know one way or the other.)
Fluorescent and incandescent lights can run off of AC directly basically because they're electrically simple (they can actually run off AC or DC without much modification) and they're designed to allow high voltages. You can take a slow-motion video of them and you'll see that they actually turn on and off as the current oscillates.
Motors like the one in your fan can also run off AC because it isn't much trouble to design a motor that can work this way.
LED lights and TVs actually convert to DC internally, using circuitry similar to the power brick for your laptop. The primary reasons your portable electronics have a separate AC-DC converter are heat being uncomfortable in something you're holding and portability, which aren't issues with the TV and LED light.
As to why designers choose to use DC instead of AC, there are many reasons. For applications like your phone and laptop that need to run off batteries; they'd need to add an inverter to convert from DC to AC that would add heat and weight to the product that is supposed to be portable. Also, essentially all digital circuitry runs off DC and it would be incredibly expensive if not impossible to design a phone or TV from scratch that only uses AC power. As with all design questions a primary motivator is cost: is it cheaper to design my motor to accept AC or to put in a transformer to convert from AC to DC?
The main reason is heat. Small devices/appliances have a harder time dissipating heat from AC to DC conversion. By placing AC/DC converter outside the chassis small devices/appliances move the source of heat away from the other components and reduce cost by not having to deal with it inside the chassis.