A degree is the scale at which you mark up waves. Every wave is made of waves of a smaller degree and is itself part of a larger one. A five on the daily chart is a single leg on the weekly and breaks into dozens of legs on the hourly.
Why it is needed
Because without a degree a wave number means nothing. "We are in wave three" is an unfinished sentence: the third inside what?
Markup without a scale can always be disputed: the same stretch decomposes differently one step up and one step down, and both decompositions are correct.
How the nesting works
By the five-three rule. An impulse divides into five waves of a smaller degree; a correction into three. Each of those waves divides the same way in turn.
Hence the thing that confuses newcomers: one and the same move is simultaneously a wave and a set of waves. Which you see depends only on the scale you are looking at.
The names of degrees
The classical scheme names the degrees — from the very largest down to the minute — with a whole system of notation: Roman and Arabic numerals, letters, brackets.
In practice the names matter less than the discipline: keep in mind which step the conversation is on, and do not hop between them mid-argument. The names themselves are a convention, not a property of the market.
How to tell which degree you are on
You cannot, in the sense that there is no absolute answer. You set the degree by choosing what counts as a significant reversal and what is noise inside a move.
A practical guide: markup is comfortable when the visible part of the chart holds several complete waves of the chosen degree. One leg on screen means you are too coarse; hundreds means too fine.
A common mistake
Mixing degrees in one conclusion. "The fifth wave ended here, but on the lower chart a third is running, so the rise continues" joins statements about different things that cannot be added together.
A fifth wave of a higher degree can perfectly well contain an unfinished third of a lower one. There is no contradiction — only a confusion of scales.
What degrees do not give you
They do not say which degree is "the main one". The coarse one is more stable and changes rarely, the fine one is more detailed and reacts faster. That is a trade-off, not a hierarchy of correctness.