Champions are elite versions of enemies and bosses. They deal and can take much more damage than their standard counterparts. They are identifiable by their bright green coloration.
They behave exactly like their standard counterparts, more care should be taken because they're much more damaging. This, along with the fact that seeing them regularly usually means the difficulty stat is very high, means they can easily take off upwards of a quarter of a maxed health bar per hit. They're also quite tanky and can take a fair bit of punishment, a fire rate boosting weapon mod and a high crit rate are very beneficial here.
The most guaranteed way to see champions is to select the nightmare perk, which sets their spawn rate to 100%.
Champions will pop up in the game from time to time. Their spawn rate increases with difficulty. Playing endless mode past round 15 will cause them to spawn regularly at about 1-3 per room. By round 27 (without Difficulty down badges) they will more or less replace the regular enemies.
At extremely high difficulty levels, Champion Bosses can be encountered. These are considerably tougher than their standard counterparts, often taking hundreds, up to a couple thousand critical hits to go down and capable of taking a huge chunk off a maxed health bar. Any champion boss that spawns additional enemies will likely only spawn more champions. Outside of the Nightmare perk, these are only likely to spawn after Endless Mode round 20 only becoming frequent after round 30.