What is "evaluation"?
Evaluation is the assessment of a position. It is similar to a human's intuition about a position (Red has an easy game, Red is better, Black is better and so on). It involves no game calculation at all, just a pure feel for how good or bad the position is. The engine's evaluation, however, computes a score for the position according to its evaluation parameters.
Before NNUE, evaluation was written by humans, who also tuned the parameters. For example, you write the most basic piece values (Chariot is worth x points, Horse is worth x points, and so on) and then tune the numbers, such as giving a Chariot 1000 points. There are also piece-position scores, piece mobility scores, scores for special patterns and so on.
NNUE completely replaced traditional hand-crafted evaluation. NNUE is like having a huge number of evaluation parameters, but humans don't know what those parameters mean.
NNUE slows down the engine's search, but since the benefits far outweigh the drawbacks (the gain from accurate evaluation far exceeds the loss from slower speed), playing strength still improves dramatically.
For a simple explanation of NNUE, see What is "NNUE"?