PvE stands for Player versus Environment. It's meant do designate any situation in which a player (or group of players) is working to overcome a difficulty presented to them by the game itself or by an AI opponent. This as opposed to PvP, Player versus Player, where they're working to defeat another human player (or another group of human players).

Any single-player game is by definition entirely PvE.

Traditional games like chess or checkers would be entirely PvP, unless played on the computer against an AI opponent.

MMOs

MMOs like Ryzom tend to be mostly focussed on PvE, usually with a few PvP activities mixed in. Hence why in MMOs the term PvE tends to be meaningful only when you mean to contrast to PvP.

Often the terms are understood to mean:

• PvP: Any activity that involves fighting characters controlled by other players.
• PvE: Any activity that involves fighting characters controlled by the AI.

But sometimes they're also used in the broader definition:

• PvP: Any activity where the challenge is to compete with and defeat other players,
• PvE: Any activity where the challenge is completely generated by the computer.

Note that situations where the players fight characters controlled by a game-master or an event team are generally considered PvE. Even if in this situation there are actual people controlling the enemy characters, those people aren't players of the game. Rather, they're game officials whose job is to provide challenges for the players, thus they're considered 'part of the game'.

It's rather futile to give specific examples of PvE activities in Ryzom, since most of the game is PvE. Usually in MMOs one lists the PvP activities and assumes the rest of the game to be PvE. In Ryzom, the most notable example of an activity specificity designed for PvP is the fighting over outposts.