Comp scoring (or composition scoring) is where you get or lose a certain number of points added to, or reducto'd from, your overall (rank-determining) score based on the composition of your army list.
Best case scenario: The TO's put out comp requirements ahead of time (ie: you start with 10 points and are allowed 2 troops plus 1 of each other FOC, -2 points for special characters, -1 for every extra FOC, -1 for every duplicate unit, etc or some other set of written rules, whatever). This system seems the worst because of how inflexible and UNFAIR it is, but at least you know what to expect.
Worst case scenario: There is a panel of judge(s) that reviews each list and assigns comp scores. This might seem like a good idea because having a set of rules like the above can cause pure Deathwing armies to score very low even though they're terrible, but having an actual person reading your list guarantees that you'll get scored appropriately. Except that it doesn't work that way irlz. The judge(s) usually use some sort of standardized criteria like the above anyways for the basis of their scoring, except they also put their own personal spin on it.
Either way is bad.
So why do people insist on comp scoring?
It promotes variety
I don't know who came up with this reason, but I imagine they had to queue into the meeting using the tele-conference equipment in their padded bedroom. How does restricting the use of some units promote variety? It discourages the use of special characters, many of which shuffle the FOC around and create diverse lists. In fact, it discourages just about anything that shifts the FOC.
Bloodwing? Can't do it (competitively) without at least 1 character (but probably more) and several troop options, netting you major comp deductions.
Loganwing? Can't do it (competitivel) without at least 1 character and basically maxed troop options. Fortunately, SW are super-customizable so it's easy to game the system and make all your squads unique, at least.
SM Bikers? Granted you technically don't need to use a character to get it, it still pretty much requires absolutely optimized maxed troops to work properly. FAIL.
Nob bikers? Ironically, this army list fits very well into the usual criteria. No characters, it can live on min troops, squads are incredibly customizable and so duplicates are easy to avoid, etc. NOTE THAT THIS IS THE EXACT TYPE OF LIST COMP SCORING IS TRYING TO GET RID OF!!!
Necrons/DE/GK/etc? Well, your codex only has 1 or 2 options for each slot so to even hope to reach the points level at all you're going to be forced into a bad comp score due to maxing your FOC. Great.
It levels the competitive playing field
Actually, no it doesn't. If anything, it makes it worse. Let's see...
Imperial Guard - Easy to game most comp systems using giant squads that only take up single FOC slots and the fact that they can take big guns everywhere.
Nob Bikers - Easy to game most comp systems using minimum troop units and extensive customization options.
Anything single-rock - Generally speaking, single-rock forces have one giant, mega-unit and everything else is just there to take objectives, or whatever. Without the need for a lot of synergy or extra support, these lists can really just max their rock and then take min of whatever else and get relatively decent comp scores while basically crushing most non-optimized lists.
Perhaps that's a bit cynical. It's easy to imagine a comp system that fails just to prove my point. I might argue that that's exactly what I just did. What can't be disputed, however, is that comp scoring causes hurt feelings worse than non-comped events for the simple reason that not everyone cares about them.
Maybe you're spiteful and bring the hardest list you can irrespective of comp scoring. Maybe you're a fluffy gamer, and your army of choice happens to be hard and score badly on comp. Maybe you only own the exact models for your single army list that happens to be hard and score badly on comp and you can't afford to change. Whatever the reason may be, some people are going to bring hardcore lists anyways.
On the other hand, some people are going to go "huzzah, a comp tourney, I can bring my fun list and not worry about anything". Or, if you'd rather me be more fair, some people will just bring normal lists. In either case, when this group of people meet the hardcore-anyways group, bad things are going to happen.
So, even being generous, some people still come with hard lists, some people still get hopelessly crushed, and variety is reduced***. Please stop with the comp already****.
* - This is actually a well-known fact, so not really a misconception. I'm getting there...
** - Necrons, arguably, don't matter since it's pretty hard to make a competitive list out of their codex regardless of comp score. Or points limit, for that matter...
*** - I can't stress enough how much this confuses and enrages me. We're going to promote variety by banning certain units. ARGH!
**** - I think comp scoring is one of those things that everyone pretty much unanimously hates, but is still popular anyways. (3D movies? Neopolitan ice cream?)
Seriously, how often do you hear about anyone enjoying comp? At best, they are indifferent, but usually it's more akin to "sigh, another comp tourney..."