Subletting
Subletting, also known as subleasing, is a rental arrangement that allows a new resident to move into your apartment (or a room in the apartment) while you move out. Whoever rents this apartment or room will pay rent and utilities, and will adhere to the lease terms. As the original resident, you're still responsible for maintaining the lease and the overall upkeep of the apartment. If the new resident flakes on the rent, you're liable for the back rent and the apartment community can sue you to get what they're owed.

You'll also be taking responsibility for all of this new renter's actions. If they throw a party and completely trashes the apartment, it's on you.

Now, because of the aforementioned downsides, not all communities will allow a subleasing arrangement because of this. In fact, most apartments prohibit subleasing agreements. If you're even considering subleasing, check to see whether or not your apartment prohibits it before you sublet. If you sublet any part of your apartment without permission at an apartment where subleasing is strictly prohibited, you'll be breaking the conditions of your contract.

Reletting?
If you want to relet your apartment, the apartment community will have the replacement resident sign a brand new lease, releasing you from all of your obligations as a renter of that apartment. If the new resident wants to throw a party and causes damage to the apartment, you won't have to worry about a thing. A reletting rental arrangement is a fresh contractual relationship between the community and the person renting out your former apartment, completely separate from the rental agreement you signed.

Reletting is your best course of action if you need to terminate your lease agreement for any reason. Your apartment community will do what it can to find a suitable renter. During their search, however, you will still be responsible for paying your portion of rent and utilities, even if you've already moved out.