Note: If you don't know what a Jungler is, see: TheOddOne's Jungle Guide
Disclaimer: Use guides to improve your understanding of something - NEVER blindly follow a guide 100% of the time, use your understanding to judge what applies to your situation and what doesn't and go from there.
Leashing is the process of drawing the aggro of jungle creeps, usually the large golem (Blue Buff) and large lizard (Red Buff) for a Jungler. Sometimes, more rarely, the large Wolf (due to the chance of high damage crits) - however, leashing the wolves can be problematic due to their high move speed, resulting in the jungler having to waste time chasing them. This is usually only done at level 1 on the first major creep the Jungler kills.
Leashing is most common from ranged champions or from champions that commonly take a ranged skill at level 1. Usually your mid/top lanes (blue side) or mid/bottom (purple side) will be expected to provide a leash if needed for the Blue Buff, and vice versa for Red. It's fairly common for a team to invade the enemy jungle early, and as such the rest of the team should usually remain close by until the minions spawn before heading to their respective lanes.
The most common form of a leash is a simple autoattack or skill from behind the wall the creep spawns by, causing the creep to start walking around the wall towards the champion that attacked them. Meanwhile the Jungler begins autoattacking/using skills on the creep with impunity. After a short period of time the creep will stop being aggro'd to the original champion and should switch to the Jungler.
It's also common for the leashing champion to continue attacking the creep and "kiting" it - staying just out of its attacking range while hitting it. This is referred to as a "hard leash" and is sometimes done if the Jungler wants to invade the enemy jungle early and needs to keep Smite for stealing an enemy creep.
Champions without a ranged attack/skill can also leash, usually by autoattacking at melee range and then immediately running towards their lane.
Note: Leashing is most effective with the least chance of a problem when done by a single person and the Jungler. Involving more champions risks the creep aggroing the wrong person at the wrong time and walking back to its spawn point and healing - this is worse for the Jungler than not having a leash at all, as it drastically slows down their jungle clear speed and tends to put them at much lower HP than they would otherwise have.
Some junglers will start at either the wolf or wraith camp - these camps spawn at 1:40, a full 15 seconds prior to the buff golem/lizard camps. A leash is not typically needed here, but usually a quick attack or two on the creeps at these camps will help your jungler out. CAUTION: Do NOT stand within experience range of these creeps when they're about to die, as taking EXP from them can prevent your Jungler from reaching level 4 on a single jungle clear (this is bad - this limits their ganking potential). Avoid last-hitting these creeps as well - again, your Jungler is working with much more limited spawns than players in lanes and every bit of gold and experience in their first clear is extremely important to them.
This helps your Jungler in several ways - they take less damage from the leashed critter while they're at their most vulnerable (level 1), allowing them to more safely clear the rest of the jungle afterwards. The small bit of extra damage helps them kill the creep a bit faster as well. The faster the Jungler finishes their initial clear (or, for level 2 ganks, the initial creep kill) the faster they can be conducting ganks. Usually there is no downside to leashing for the laning champion beyond a small mana/energy cost - there's no reason a proper leash can't be completed with the champion back in their lane prior to the first minion dying.