How do you prevent Sims from aging?

Making a Sim "Younger"

Life Fruit

You can reduce a Sim's age by a single day by eating a raw Life Fruit. However, they can only be planted by a level 7 Gardener and the "Unknown Special Seeds" or "Life Seeds" are only available under certain circumstances:

Sims can receive the seed by searching the town, as a science career reward, by exploring the catacombs in a mausoleum, fishing it out of a body of water, or while using the Time Machine.

_{Source: Sims 3 Wikia, Life Fruit page}

Ambrosia

The better option is to cook Ambrosia, which resets the Sims age to a predefined value (adults eating it will reset to have 26 days until aging). It can also bring Ghost Sims back to life.

Ambrosia requires purchasing the recipe (§12000), a level 10 Cooking Skill and then one Life Fruit and one Deathfish as ingredients.

A deathfish can only be caught with level 10 Fishing skill, from the Graveyard between midnight and 5AM and needs angelfish bait. Note that if you catch enough deathfish you can stock a pond and have a supply of these available all day round.

Young Again potion

This potion requires the Generations pack and is created using a chemistry set, it resets a Sim who is older than a young adult back to day 1 of the young adult.

You will need a high Logic skill for this.

Prevent / slow down further aging

Age Freeze Potion

Another potion, so also requires the Generations pack. This is a lifetime reward that requires 65000 lifetime happiness to obtain. Drinking it will completely stop your Sim from aging further. Note that if you eat birthday cake after drinking the potion, your aging will start up again.

Change form

If you have the relevant expansion pack, you could also get the Sim converted to a Vampire or Mummy, as they have longer life spans.

Cheats

If you enable the "god mode" cheat (testingcheatsenabled true) in the cheat console, which is activated by pressing control, shift and c in unison, you can decrease the sim's age, but only in the given stage of life. By shift clicking the sim, and editing them (you don't even have to change anything) in create a sim, the game will set their life back to the first day of whatever stage they are in.

You can stop (not reverse) the aging process via the general menu.

There is a checkbox for "Enable Aging".

You can adjust the lifespan.

Here's a screenshot:

Look under Options -> Game Options for the option 'No aging'.