You can grow any type of vegetable in a container, (depends on your definition of a container!), but if you are referring to smaller pots, troughs, grow bags etc then all sorts of vegetables can be grown, aubergines, peppers, beans, peas, tomatoes, strawberries, rhubarb, salad crops etc.
The main advantages are no poor soil problems, no weeding, no digging and no soil pests! You can also move your plants around in pots to take advantage of the sun or move when bad weather is imminent......good for tender veg as well.
The main draw backs of container gardening will be smaller harvests, and more regular watering and feeding, so only grow something that you really want to have a go at or if you have no choice, e.g. you only have a balcony to garden on, then it is your only option, but one I would still pursue if it were me in that same position!!!