1) Elo Rating

Its main Advantage is surely that it measures skill very accurate and that it doesn't take too long to find your appropriate ranking. On the other hand it can be kind of demotivating as you start with an average rating (not at the bottom like in others) and lose as much points as you win when playing vs an equal rated opponent from the first game on. With Elo roughly 50% of the players end up with a score lower than the score they had when they first enter the Realm of Multiplayer. Elo also doesn't work reliable for Random-Teams, just for 1vs1 and "Arranged Teams" where the Team as fixed entity with always the same players has 1 unique Elo-Score. Usually Elo-Rankings never get reseted. 

2) Systems à la Warcraft III

In Warcraft III you start at level 1 and gain XP for each win and lose XP for each loss, however until you reach a certain level you get more XP for a win than you lose for a defeat, so every player will rise at least a few levels before he reaches a level where he can't rise until he becomes better. Basically  you don't start in the mid of the bell curve like you do with Elo but on the very left
So even the worst players will get to level 10 (or so) if they just play enough. To quote Blizzard on it :

The most highly skilled players are expected to be level 50; players of average skill are expected to be around level 30; and even the least experienced players of Warcraft III are expected to be level 10

I know I repeat myself but with this system every one is a winner when he plays ladder - rising to Level 10 is a lot cooler than starting at 1600 and falling down to 1400 - especially considering that you are probably better than you have been when you still were at 1600.

This surely is more motivating for the players than Elo-system, on the other hand you need to play very much games to get a rating equaling your skill level.  I don't know the details of the system but aside from your actual level you also have a "expected ladder level" (ELL) which is the assumed level you will reach with your current skill when you played enough games. This ELL determines  who you get matched up with and, in combination with the actual level of you and your enemy, how much points you get
Additionaly you lose points if you don't play enough games per week (up to level 10 you don't lose points, when level 30 you need to play 10 games a week or so to not lose points at very high levels you need to play more than 25 games a week to not lose points) 

The Ranking gets reseted every now and then, ranging from 6 months to more than a year I think. 


3)Ladderstyle means that it is a positional system (there is no rating which defines your position, just the position itself) with rules like: if you win vs a higher ranked player you rise in the ladder by a certain amount of positions (for example half the difference of your positions, or you get his positions or 10 positions or half the difference but max 10 positions), if you win as higher rated player you get 0 or 1 positions up usually, if you lose as higher rated player you lose 1 position usually, if you lose as lower rated player nothing happens or you lose 1 position usually. There are a lot other possibilites what happens but I think thats the most common. There are special rules for top 10 or top 5 and how to get #1 usually. 
Usually you drop in the ladder for inactivity / refusing challenges of people near your position 


4) ICCUP is a big private starcraft league ( I think one most active esport leagues worldwide with about 150.000 Games played per week), which uses the following quite simple system: 

you start at "D" with 1000 Points. for each win vs another "D" guy you get 100 Points for a defeat you lose 50 points. If you play vs someone higher rated (i.e. D+ or C-...) you win more and lose less. 

If you fall below 800 points you are "D-" if you get to 2000 Points you become "D+". 
when you have 3000 Points you become "C-" and now you lose 75 Points per game. 4000 = C; 5000 = C+ 
when you have 6000 Points you are B- and now you lose 100 Points so to rise any further you need to win as much matches as you lose. 
at 9000 Points you get A- and now you lose 125 points for a defeat, so only the top of the top can rise any further. 

(on ICCUP you get 130% points if you play on the "maps of the week", so it is a bit easier to rise than I outline before, but as Demigod will have automatches on random maps I guess, Maps of the weeks aren't suitable for Demigod, but maybe Tournament games could give that bonus points) 

a season on ICCUP usually lasts 3 Months. 

My Personal Favourite would be a ICCUP-style system complemented with a Elo-Rating for a all-time record. (but the ICCUP-style rating would be what is necessary to qualify to the season finals)

I have yet to dig my head into Microsofts "Trueskill" system but they claim to be able to rank random teams reliable with it, so if anyone knows details about it, feel free to add it.