Learn French: On ne peut pas conduire sans permis valable. - We cannot drive without a valid license.

In French, when you use sans (without) followed by a noun, you usually drop the indefinite article (un, une, des) if you are talking about something in a general sense. So, "without a license" is simply sans permis.

The translation says "We cannot drive", but the French uses on. Why not nous?

On literally means "one" (as in "one cannot drive"), but in spoken French, it is the most common way to say "we". It is also used to make general statements like this one, meaning "people in general cannot drive". Remember that on always takes the same verb form as il and elle.

Does the 's' at the end of permis mean it is plural?

No, un permis (a license or permit) naturally ends in an 's' even in the singular form. If you needed to make it plural, the spelling would stay exactly the same: des permis.

Why is conduire left in its full infinitive form here?

Because it directly follows the conjugated verb peut (can / is able to). In French, when a verb directly follows a conjugated verb, the second one stays in the infinitive (the dictionary form).