There is an excellent guide to MRI at mriquestions.com. Most images in these notes come from that site, courtesy of Allen D. Elster, MRIquestions.com.

How are T1 and T2 determined?

Assume, for now, that the subject being scanned is of a single substance with the same T1 and T2 everywhere.

How to find the different T1 and T2 in different locations will be discussed later.

Discovered 1949 by Erwin Hahn ... Probably the most important contribution to MRI.

The signal (which is picked up by the transverse coil in the $xy$ plane) provides T2*, not T2. T2* is mainly caused by dephasing due to field inhomogeneities.

Adding another pulse to "flip" the magnetic moment by 180 degrees causes the individual to get back into phase, providing a strong signal at twice the pulse time.

The time of the echo can be used to compute T2, but really provides a greyscale value in the MR image with intensity propotional to the echo magnitude.

More details are available on the Wikipedia page.

A gradient in $B_0$ is a variation in $B_0$'s strength with position.

Initially, $B_0$ is the same everywhere. But "gradient coils" can be activated to cause $B_0$ to vary with position. Here are some typical gradient coils:

Important note: The direction of $B_0$ is not changed; only its strength is changed.

This shows the strength and direction of $B_0$ with arrows:

We can also produce echoes in the T2* signal (not T2) by applying a gradient field to the subject.

The gradient causes MR with different strengths at different positions, so the frequencies of the precession change and things get out of phase more quickly.

A negative of the same field for the same time reverses the freqency changes, so the precessions move back into phase.