Let's avoid a debate over semantics; I will define what I mean by granularity up front. If you want to use another word for it, be my guest.
Granularity is the measurement of the size of scoring increments in a game. It can be constant from start to finish, or change as the game progresses. It is a concept that is not often discussed along with the usual theme/mechanics aspects when critiquing a game.
I tend to like games with either constant granularity, or games in which you have quite a bit of control over the sizes of the scores and can make decisions based on risk/reward. In the former, the scores you get each turn or each scoring event are fairly close each time. This can be 1 point or 10 points, as long as each time you score for something, it's relatively the same value. In the latter, the scores can change, but you have some control over when you get them, and may choose to allow others to get them to improve your position.
Examples of fairly constant granularity are:
Yinsh: 1 point for mking a row of 5. The only way to make more than 1 point is to form crossings where one row of 5 doesn't overlap any discs in another row.
Euphrates & Tigris: Scores increase 1 VP for laying a tile (usually) and winning internal conflicts. External conflicts are often a bit more but, in my experience, tend to be in the 1-4 range.
Hansa: You get 2-4 points per good sold based on the number of barrels, and 1 point for each unsold good.
Examples of changing (but controlled) scoring:
St. Petersburg: Money increases as the game progresses. This is used to buy more money- and VP-generating cards. All players follow a similar progression. If a player falls behind in VP somewhat, they may be putting themselves in a position to buy something worth more VP in the long run.
Carolus Magnus: Each turn, you may only place 1 tower maximum. However, when you take over an opponent's territory, you replace all his/her towers with yours. This can make the game go from 8-5 to 4-9, for example. You have control over reinforcing something valuable (many towers), but may choose to risk it to steal/maintain control in the court, etc. This game is about at the extreme level of granularity for my tastes; it is so damn elegant.
Carcassonne: All versions. There is a fairly constant availability of all manner of scores throughout the game: 2-point cities, 3-point roads, 9-point cloisters, 12-point cities. There are also some very high-scoring areas (large cities and fields) that, to some extent, all players have a chance to fight for. If you choose not to, or are unable, there are ways to mitigate the loss.
Uncontrolled granularity, in my opinion, is where many games fall apart. Players fight for 1- and-2 point scores the whole game. Then BAM! One player gets 50 points. Not fun. The awards organizations seem to love giving props to games where luck of the roll/draw/guess counts more than skill.
Examples of this are:
Ticket To Ride: If you make a 20-point ticket, you get 20 points. Otherwise, you lose 20 points. What the %$@# kind of rule is that? Well, this game won some fancy schmancy award, has some nice quality bits, and it grabs the attention on non-gamers for being a pretend-train-game, but puh-lease.
Niagara: I might say this was more a chaos issue than granularity, but they are related here. Although you score for dropping gems at the base 1 at a time, through no fault of your own, you can go from 2-2-2 (1 in your boat) to 0-3-4 because 1 gem was stolen and one of your boats went over the falls. A 4-point swing may seem small, but in this game it can mean everything. You can argue that you have control over all this, but it just isn't so. Thankfully, BSW implemented this game. Now I know it's awful and will not buy it.