Standard Uniform Distribution

Never really did much stats so this is probably an easy question.

What values can the standard uniform distribution take? Is it just any value between 0 and 1? Is each value chosen with equal probability?

So for example, if I'm to asked the question...

Calculate $\displaystyle t_1$ ~ $\displaystyle U(0,1)$ where U is the uniform distribution and t is a random variable. Is that even a properly written out question? Does it just mean t will take a random value between 0 and 1?

As I say I've not done much stats and not entirely sure what the notation stands for.