Ego possum dormire in villa, sed tu potes vigilare in horto.

Yes. In Latin, the verb ending usually already tells you the subject:

• possum = I can
• potes = you can

So ego and tu are not strictly necessary. The sentence could simply be:

Possum dormire in villa, sed potes vigilare in horto.

When Latin includes ego or tu, it often adds emphasis or contrast. Here, the contrast is very natural:

• I can sleep in the house,
• but you can stay awake in the garden.

So ego ... sed tu ... helps highlight the difference between the two people.

They are forms of the verb posse, which means to be able or can.

In this sentence:

• possum = I am able / I can
• potes = you are able / you can

So Latin is using the same idea English uses with can, but grammatically it comes from a verb meaning to be able.