Demesne is the total amount of holdings that you, as a ruler, have under your direct control. The holdings that you own personally will always have your banner on the right and left side of the picture when you click on a county.
Your demesne limit is displayed in the upper right corner, for example 12/7 - the first number is how many you control directly, the second number is what your limit is. In this case, the demesne limit is exceeded by 5 holdings. Controlling more holdings than your limit allows will give you negative tax modifiers for every holding above that limit. Having a demesne over the limit is not always bad, though. It is sometimes more beneficial to hold all the best provinces yourself even if your vassals think slightly less of you for doing so.
For every holding you control over your demesne limit your vassals' opinion of you will drop by 10. Hovewer, it is always good to control counties directly since you can upgrade structures on your own, raise your own levy, and get direct income. Choose wisely - give away weak and under-developed counties to your vassals, but keep the good ones to yourself.
Increasing your demesne size (the list is not complete, please add and delete this sentence)
Increasing the limit is difficult, as the game revolves around gaining power but maintaining stability, however it is possible. Your personal stewardship, added by half of your wife's stweardship stat, will increase your demesne limit. The bonus is calculated in this way : (your stewardship + spouse stewartship/2)*0.15 and is rounded down. As such, finding a wife with a high stewardship is vital for trying to rule many holdings, and a important factor in choosing a Marriage.
Based on your title, you get a bonus to your demesne size limit: emperor 4, king 3 and lower titles 1. In addition, a Duke in control of multiple duchies will get the "Great Duke" bonus, increasing his demesne limit by 1. If your ruler has the Gavelkind Succession Law, the demesne limit will be increased by 30%.
Researching "legalism" technology will further raise the limit by 1 for each level, though this will increase slowly and over centuries. Overall, Crusader Kings 2 does not give too much space to control all your lands and conquest spoils directly. It is not meant for you to control all the provinces personaly, the beauty is that you rather have to pay much more attention to those whom you give the land to. As years pass by, your ruler will become stronger and stronger, able to control more and more, just as the age of feudalism comes to an end, opening the door for strong absolute monarchies.
Effects of going over your demesne cap
The effect starts out at -10 vassal opinion per holding if you go over the cap. This effect gets rather dangerous if you push your overextension too much. If you have counts and higher as your vassals, they will start forming factions against you. These will start with quite fair demands such as lowering laws, but will soon develop into revolts for their own land. There is however a way to play around with your demesne: remain a count.
(<--- ! dude check comments! being over demesne size decrease demesne levies and tax income by a large amount and you wont have any research: thats a pretty bad idea..)
You will notice that despite that your mayor and bishop vassals will start to like you less, they won't be able to revolt or even form factions. However, if your overextension reaches around 400% other rulers from outside your realm will go crazy for your ground. You'll have a constant war for your lands, until your demesne is reduced again. (Aka, until you form the duchies and counties out of it)