Here’s the punchline: tempo runs dictate your race performance.
When you’re training for anything over 10k, and you’re past the point of just surviving and want to actually crush it, tempo runs are your best friend. And probably your worst enemy. You may have found that folks are throwing the phrase “tempo run” around quite a bit, but my favorite definition from that article (and the one I’ll use here) is as follows:
One fourth to one third of race distance at race pace.
So to use my own situation, in which I’m training for a half-marathon and want to finish in under 1:40, I need to be semi-regularly running 4-5 miles at 7:38 min/mi pace. Though realistically, my tempo runs need to look like this:
The whole point of tempo runs is to get your body acclimated to racing. Long runs are great because your body needs to get used to the distance; easy runs are great because you need to rest; speed work is great to push your body’s anaerobic limits and increase your V02 max. But at the end of the day, tempo runs are what carry you through a PR on race day.
Tempo runs are supposed to closely mimic your race environment: you’re pushing a hard pace for a handful of miles (just like a race) without resting in between (just like a race). It gives your body and your mind a taste of how you’ll handle the rigors of a full race. This is why speed work, in my opinion, is less essential to training than tempo runs: you don’t get to rest in between miles during a race! Plus you certainly won’t be pushing as hard as you can for one mile at a time before taking said rest.
One thing (of the many) I learned from my years in team sports: you practice like you play. You have to be put in game-day situations during practice, so when those situations happen during the game, you’ll know what to do and can react without thinking. Whatever your race strategy is, you have to practice that exact strategy while training for the race. Tempo runs are the next best thing to actually racing.
As always, Runner’s World has a great write-up on what tempo runs are, why they’re so beneficial, and how to go about implementing them correctly. There’s a definite reason why these, to me, are more intimidating than long runs or even track workouts, and therefore even more rewarding. If you already have a solid mileage base and are looking to improve your current race times, tempo runs will get you there.
Where are the pancakes? 
I would add that tempo pace, for those of us training for a half-marathon and beyond, is done at about 10K pace (maybe a hair slower). My half-marathon goal pace is challenging over 13 miles, but a pretty sustainable cruising pace when I’m “only” sustaining it for anywhere between 3 and 6 miles (or whatever). So I run FASTER than half-marathon pace, a much tougher basically 10K pace, to simulate race-day pain and get that cruising half-marathon pace feeling easier come race day.
[preaching to the choir, I know. Consider this a post-script or footnote to your post]
That’s a good point. In that respect, the quote from RW at the start of my post is a little misleading. If you’re aiming to run 13.1 in some amount of time, tempo runs that aren’t 13.1 should be faster than your goal time.
Unless, of course, we ever end up doing a 13-mile tempo run 🙂