ездить - to drive, to ride, to travel, to go (by a vehicle) - this is the imperfective miltidirectional form (as you may know, motion verbs in Russian have a multidirectional form, and a unidirectional form)
Examples:
Я часто езжу домой, к родителям. (round trip)
Он часто ездит на своей машине. (general motion, no direction implied)
ехать - the same meaning, also imperfective, but unidirectional movement, for example:
Куда ты едешь?
После работы я всегда еду домой. (one way movement, always from work to home)
поехать - perfective form of ехать, meaning completed action; also has the meaning "to leave (for), to start moving"
Examples:
Он вчера поехал в отпуск. - He went on holidays yesterday. (= he left town)
Из-за закрытия воздушного пространства, они поехали в Германию на машине. - ... they went to Germany by car (like in English, this can mean "they left", but can also refer to the completed action)
поехали! - let's go! - not necessarily by some vehicle, this can also be used as a general expression for "let's start (some activity)"
доехать - to reach, to arrive, (perfective); the imperfective form is доезжать
Examples:
Новый российский посол наконец доехал до Киева. - The new Russian ambassador finally arrived in Kiev.
Он доедет туда за полчаса - He'll get there in half an hour
Как вы доехали? - Did you have a nice journey?
До работы доезжаю минут за 40 и утром грею минут15, приезжаю, а в машине все равно прохладно. - It takes me 40 minutes to drive to work, in the morning I warm it up for 15 minutes, but when I get there, it's still cold inside the car. (See another prefixed verb here: "приехать, приезжать", which means "to arrive", while "доехать,доезжать" is closer in meaning to the English verb "to reach")