Power leveling is probably the term you're looking for. Wikipedia description.

There's many ways to help someone gets to the highest level, one of which is to give them better items/equipment they wouldn't have access to when starting, twinking, which is best described in GodEmporerDune's answer.

Another is to kill and/or assist them in killing to get more experience then they would alone.

Doing both is sometime counter productive as, often in games, items have a level requirement or statistics that are level based. It's preferable to wait till the new character is at the highest level to give them items.

The most common way to power level is to have a high/highest level character do all the work while the lowest level character reap the rewards, usually by being in the same party.

For example, in Everquest, a way to power level would be done with healing-able classes (Clerics, druids, paladins, etc) which would heal and buff the lower level character while they fought higher level monster that they wouldn't normally be able to fight alone. This and twinking the lowest character was a way to power level.

In WoW, this thread shows ways to power level that are quite different. Wasn't like that when I was playing but the ideas are similar, have the highest level characters do the work while the lowest gain experience. It was also possible to do it like it was done in EQ, high level character buffs and heals the lower level one.

In Diablo 3, one power level technique, with RoS, is to have a level 1 character starts a game in the highest torment level they can, invite a level 70 character and do rifts. The 70's character open the rift, clean the entrance, the level 1 enters the safe, cleared, zone and waits there while the 70 clean the rift level. The level 1 character gets experience with the kills and with the end of the rift.

Power level guides can be found for many many games because it ain't the same for all games.

GodEmperorDune's excellent answer provided some of the most common terms. Below I'll list a few less common terms as an addition to his answer.

Supporting with high-level buffs/heals: Babysitting or Nursing

High-level magic users can provide healing or supporting magic (buffs) to lower level characters. While this is often done as part of power-leveling, it's not always just to gain EXP. A common example is in PvP.

For example, in the MMORPG EverQuest, the Rallos Zek server allowed players to attack anyone at any time at any location. The one restriction was that players had to be within +/- 4 levels to attack another. High-level magic users couldn't directly fight in low-level battles, but they could babysit low-level combatants through buffing and healing.

In some cases, high-level characters could intervene in low-level battles by "pulling trains". The high-level character would aggro a lot of mid-level monsters, then bring those monsters storming through a PvP combat scene. Intentionally forming such a train of monsters is often a prohibited and bannable offense, but this doesn't stop it from happening.. or even being popular: Leeroy Jenkins! If a healer or teleporter, the high-level character could then save low-level characters at will while leaving the monsters to finish off the rest.

Sharing high-level travel methods: Transporting

High-level characters have access to advanced forms of travel. Common examples include teleportation magic, being keyed into certain zones, share-able mounts, money to pay for convenient NPC-provided taxi services, and travel-enhancing magic (e.g. buffs that make characters run faster).

Low-level characters in large worlds often want to be transported to their favored starting ground so that they're not stuck with their default race/class starting area. Also, many low-level quests used to be based on traveling across the world (which was hard for low levels in some MMORPG's), so transporting helped low levels get quests done.

Directly assisting with quests: Multi-questing (MQ'ing)

Some MMORPG's allow items for collection quests to be traded. This item-sharing process is called multi-questing. High-level players can collect difficult-to-get items and give them to low-level characters who in turn complete quests otherwise inaccessible or impractical for them.

A very common multi-quest item was the Ring of Ancients in EverQuest. A very rare spawn would rarely drop a ring. When turned in, the character completing the quest would get non-tradable boots that would provide a significant boost to their run speed.

Assisting with a reputation grind: Factioning (faccing)

MMORPG's like WoW and EverQuest include a faction system in which certain groups of NPC's had their friendliness depend on the character's faction score. Faction scores can often be increased by repeatedly killing (grinding on) monsters that the faction hates. High-level characters used to help low-level characters by slaughtering faction-yielding NPC's while the low-level character is in group.

In some cases, the low-level character would need to be considered a combatant to get a faction hit (or rep increase). In these cases, the low-level character could provide token support such as by healing the high-level character or using ranged attacks.

Providing social support to new players: Cheerleading

Just like in real life, established players can provide social networking support for new players in social organizations. Whether it's a mentor helping you at work or a high-level player getting you into a good guild at Level 5, they're your "cheerleader".