Мы с тобой: you and I
Мы (we): first person plural pronoun, used for referring to yourself together with other people. Straightforward, right?
Well, sort of. Мы means "we," but it can also mean "you," "him," "them" or just about anyone else, abstractly and specifically. It's one odd little pronoun.
First oddity: When мы (we) means я (I).
In English, we like to distinguish the individual participants who make up "we": George and I went to the movies. In Russian, this kind of movie-going lands you in the мы zone: Мы с Гришей пошли в кино (literally, "We went to the movies with Grisha," which you say when it was just the two of you). It's hard for English speakers to get used to this, but it's worth the effort. Мы с тобой уже видели этот фильм. (You and I already saw that film.) Saying "Гриша и я пошли в кино" or "Ты и я уже видели этот фильм" is like branding FOREIGNER on your forehead. And it sounds wacky, too.
Another case of мы = я is the so-called авторское мы (editorial we), which is also charmingly called формула скромности (the modesty formula). This is the convention of using мы in scientific and scholarly articles as well as newspaper editorials even though it was just one writer, scientist, etc. English speakers do this, too: В предыдущих лекциях мы рассмотрели, что такое статистические расчёты (In the previous lectures, we considered statistical calculations).
And then there's a мы = я usage that is not used colloquially but fun to know anyway: Мы, Император всероссийский … (We, the Emperor of all Russia … ). I have read that this was a reference to divine right — "God and I declare" — but it may be a ceremonial way of speaking not just on behalf of oneself, but on behalf of the nation.
Moving along, sometimes мы = все (everyone) is an abstract group of people. Мы все умрём (We will all die). Вместе мы изменим мир (Together we can change the world.)
In certain contexts, мы = ты/вы (you). In hospitals, schools and hair salons, doctors, nurses, teachers and beauticians often use мы when they mean ты. In the literature, this is presented as a way of forming a bond with another person, taking part in their life. But most people find this usage to be incredibly patronizing and annoying. Как мы себя чувствуем? (How do we feel?) Открываем наши тетради (We open our notebooks). Что мы сегодня сделаем — только стрижку? (What are we doing today — just a haircut?)
Television cooks do it, too: Режем лук и жарим его на постном масле (We cut up the onion and saute it in vegetable oil).
And then in the dog park мы = моя собака (my dog). This seems to reflect the pet owner's total identification with his or her pet. Or maybe the owner is speaking for the dog. In any case, you will hear the rather saccharine Мы боимся больших собак (We're afraid of big dogs), or the sickly sweet Мы покакали (We pooped). It's the dog talking.
Finally, there is the comical usage of мы = я. Let's say someone is trying to sweet-talk you. You reply: Знаем мы вас! Не пройдёт! (We know your type. It won't work.)
Who is мы? Why is this funny? I don't know. But we are very amused.