Upkeep is the maintenance cost for certain permanent skill effects. Currently, the only skills requiring upkeep are some Monk and Assassin Enchantments; and the only kind of upkeep is "-1 Energy Recovery". Effects that require upkeep can at any time be canceled by the caster, by double-clicking the icon in the upkeep bar, which makes both the effect and the upkeep end.
When maintaining skills that have upkeep, the caster needs to be mindful of their reduced energy supply or try to get more energy some other way. For example, Mesmers can utilize Inspiration Magic that will increase their energy reserves, and a Warrior could use mostly adrenaline-based skills. While most characters will be content using one maintained enchantment (if any at all), Monks often maintain many of them and regain the required significant amounts of energy through Blessed Signet and/or Balthazar's Spirit. See Bonder for details.
If a player has negative energy regeneration and the player's energy would drop below zero, the player loses maintained enchantments until their energy regeneration reaches 0, starting with the last enchantment cast. For example, if a Monk were maintaining 8 enchantments (putting their energy regeneration at -4 pips), they would lose four enchantments, setting their energy regeneration to 0. This is especially common (and devastating) in the case of a bonder.
There is a hard maximum to the number of enchantments you can maintain: You may never have more than 10 energy degeneration. (This translates into 14 enchantments for a character with 4 pips of energy regeneration, and skillful bonders may reach this mark.) When trying to add another maintained skill to the upkeep, oldest one is canceled and replaced by the new one. The most important consequence of this cap is the interaction with Nature's Renewal, which doubles the upkeep cost and thus halves the maximum number of enchantments that can be maintained. You can also get additional energy regeneration and thus the option to sustain more enchantments through skills like Blood is Power, Blood Ritual, and Succor.
Note: There are Hexes which incorporate energy degeneration, such as Malaise. These are not considered upkeep because they do not last forever and don't offer the same canceling interface as upkeep enchantments do. They too will be lost upon the death of the caster, or often if the target's energy drops to zero.