A pinch pot is the simplest (and likely oldest) method of making a clay pot. Take a lump of nice pliable mud, roll it into a ball, use your thumb or a finger to push a hole in it, and shape it using your fingers to open the inside and your other hand to support the outside, smoothing and forming the bottom and walls until it's a shape and thickness you like. This is best used to make relatively small pieces; anything much bigger than your hand will get unwieldy, and it will be easier to build it out of coils or slabs.
That shape doesn't have to be round. Wheel-thrown pots default to being round, although it's possible to reshape them afterwards: I've seen videos of people making oval baking dishes by throwing a round pot and then stretching it into an oval while it's still plastic, and there's a really cool Carthaginian oil lamp that's clearly a thrown pot with the sides pinched in to make three spouts. I want to try replicating that one when I get a chance. A pinch pot can start out any shape, although I'd avoid sharp, square angles; those are easier achieved building with slabs.
Not having access to my usual weekly pottery class, I've been playing with some hand-building techniques while I'm home, including pinch pots. So far, I've made two oil lamps: since it's easy to make an eccentric shape this way, these are teardrop-shaped vessels about the size of my hand, with a spout for the wick at the narrow end and a fill hole in the top. I make the top and bottom as separate pieces and stick them together using the same technique I learned to attach a handle to mug: scratch up both surfaces with a sharp tool (or a toothpick, whatever is handy), paint them with vinegar, repeat, then stick the pieces together, compressing and smoothing the join with my fingers. Once the piece has dried to leather-hard - the clay is no longer squidgy, but hasn't gone chalky - I smooth down the outside and carve any decoration I want into it.
One of the reasons I like pottery so much is that it's a very tactile art in both the making and the result. Potters use all kinds of tools, but in the end it's mostly hands shaping clay, and the finished pots can be as smooth as porcelain or as rough as a grindstone, burnished smooth or carved, glazed or not. Pinch pots are even more so: all the shaping happens between the potter's hands and fingers.