Why a second steep is often better
The first infusion extracts the most soluble compounds fastest, which includes a lot of the bitterness. By the second, the leaf is fully open and giving up its aromatics more evenly. Many oolongs and puerhs are widely considered to peak at the second or third infusion rather than the first, and a tea judged on one steep is a tea judged before it started.
How many times
Depends on the leaf and the style. Broken-leaf black tea in a mug: once, maybe twice. Whole-leaf black or green in a Western pot: two or three. Rolled oolong or compressed dark tea gongfu style: six to ten is normal and more is not unusual. The leaf tells you when it is finished — the liquor pales and the aroma thins.
The technique that matters
Pour the vessel out completely each time. Leaf sitting in residual liquid keeps extracting between infusions and the next steep arrives bitter through no fault of your timing. Then increase the steep time as you go: if the first was thirty seconds, the fourth might be a minute. The leaf gives up less each time and you compensate with duration.
Between sessions
Wet leaves left for hours are a quality question rather than anything else — they lose aroma quickly and the resulting cup is flat. If you must pause, a few hours at room temperature is common practice in many tea cultures; overnight generally is not worth it for flavour.