(Yes, I’m back to writing!)
When fitting a ship in Eve, you generally pick a propulsion module that matches the size of your ship — frigates and destroyers use 1MN afterburners and microwarpdrives, cruisers and battlecruisers use 10MN modules, and battleships use 100MN modules. With one exception , you should never use a prop mod that’s smaller than intended for the hull.
However, there are occasionally good reasons to use a propulsion module that’s larger than intended, especially afterburners. (It’s particularly popular right now to use 10MN afterburners on the Tech-3 destroyers, the Svipul and Confessor.) Why would you do this?
First, an oversized afterburner allows you to move very fast, as long as you don’t have to make any turns. This can be surprising to newer players; if you look at the stats for a 10MN afterburner, it lists a 135% maximum velocity increase, compared to the 500% increase on a 1MN microwarpdrive. However, they have different thrust amounts!
The actual top speed given by a prop mod is actually proportional to both the maximum velocity bonus, and the thrust generated:
Vmax = ship base speed * (prop mod boost% * (prop mod thrust / ship mass))
So, for most frigates and destroyers, overheating a 10MN AB will give you a top speed equivalent to a non-overheated 1MN MWD! 
Secondly, using an AB allows you to ignore warp scramblers. Warp scramblers have two functions: they prevent a target from warping away, and they disable any MWDs on the target ship, reducing their ability to maneuver.
One way to keep mobile while scrammed is to fit both a MWD and an AB, and switch to the AB once you’re scrambled. (This is commonly called a “dual-prop” fit.) However, a properly sized AB won’t give you the same speed as the MWD — plus, it takes up a second mid slot, which most ships don’t have many of. Another solution is to use an oversized AB as your sole prop mod, which gives you speed similar to an MWD in a single mid slot.
Keeping mobile while scrammed is particularly useful for the Svipul; it doesn’t get a turret range bonus, so it almost always operates at ranges of 15km or less. (Confessors get a range bonus while in Sharpshooter mode, giving them a little more flexibility.)
Finally, ABs of any size do not have the “sig bloom” of an active MWD. If two ships with an MWD and a 10MN AB are moving at the same speed, the AB ship will take far less damage from missiles, and be hit less often (and for less amounts) by turrets. Thus, oversized ABs can act as a form of damage mitigation.
However, there’s also some significant downsides to using an oversized AB:
- Every propulsion module has significant mass, which is added to your ship, similar to an armor plate. The 10MN AB is nearly four times as massive as an entire Svipul! While an oversized AB allows you to move at high speed in a straight line, it robs you of your agility. You have a massive increase in inertia when you use an oversized propulsion mod; you can’t turn on a dime, or hold a tight orbit, or sling-shot a long range tackler.
- Due to the increased mass, oversized afterburners also produce less acceleration than a MWD. Even though they have similar top speeds in a straight line, an oversized AB will take 15-20 seconds to reach that speed, compared to 3-5 seconds for a properly sized MWD. This is particularly painful if you jump into a gatecamp and need to reapproach the gate.
- An oversized afterburner requires nearly three times the power grid of a properly-sized MWD to online. This usually means compromising some other aspect of your fit: giving up tank, firepower, or utility modules.
One of these downsides limits the number of ships you can use this on. Because you can’t turn quickly, it’s nearly impossible to manage your angular velocity in a ship with an oversized prop mod; so, you generally don’t want to use them on turret ships. Save oversized afterburners for ships with bonuses to drones or missiles.
Also, this technique isn’t limited to frigates and destroyers; there are a number of cruisers that perform well with 100MN ABs fitted instead of 10MN MWDs. (The main candidates here are drone cruisers such as the Gila and Ishtar, although it can work well for missile cruisers as well.)
Update: Added a mention on sig bloom mechanics, per some smart eyes on Reddit.
1: Supercapital ships will often carry an X-type 100MN MWD. It won’t actually allow them to move quickly in a straight line, due to the monumental mass of their ship; however, they can use it to speed up alignment times when warping to celestials, or to use it as an E-brake to prevent being bumped out of a POS.