These new electrons (or holes) can come from many sources: electrostatic tuning, charged impurity atoms, adsorbates etc. This condition of excess carriers, which corresponds, as you mentioned, to the Fermi level being away from the Dirac point, is known called doping. The term is somewhat misleading since we conventionally talk about doping as introduction of donor/acceptor atoms in semiconductors. In graphene, however, doping is commonly referred to in a broader context.
Excluding electrostatic tuning doping normally has to do with some kind of disorder in the system. This disorder, most of the times, is responsible for doping in the conventional sense, i.e. accepting or donating electrons. But in the case of electrostatic tuning the contacts facilitate the injection of electrons or holes in the graphene sheet. As an example, consider the back-gated graphene Field Effect Transistor (gFET) in Figure 2 (a) (on page 4/10) in:
http://journal.insciences.org/wp-content/files_mf/1664_171x_1_2_80.pdf
All voltages mentioned below are measured with respect to the source (considered ground). The drain voltage (##V_d##), or drain-source voltage, is typically in the tens of millivolts, which in turn gives a current of couple of microamperes during transport. You can figure out the exact numbers from part (b) of the figure. For the purposes of this discussion assume ##V_d = 0##, and the gate voltage ##V_g##, or gate-source voltage, is 10 V. Consequently, the contacts, which are connected to your voltage source, are negatively charged and the gate is positively charged, i.e. like a parallel plate capacitor. Remember that since graphene is conductive under all conditions (i.e. no band gap) it acts like a metal and it also gets negatively charged by accepting the excess electrons from the contacts, which in turn get replenished by the voltage source. Therefore, the contacts and graphene combined act as the negative plate of a parallel plate capacitor, with the gate acting as a positive plate.
After quoting Wikipedia
In a semimetal, the bottom of the conduction band is typically situated in a different part of momentum space (at a different k-vector) than the top of the valence band.
it is obvious that one big difference in graphene and a "typical" semimetal is that in graphene the conduction and valance bands meet at the same point in k-space. Here is the link to the Wikipedia article:
http://en.wikipedia.org/wiki/Semimetal
If you look at part (c) of the figure then another important difference can be observed. Since the Fermi level crosses the conduction and valence bands you can have holes and electrons existing in the semimetal
simultaneously. Whereas in graphene you can have electron conduction in one condition and hole conduction in another, but not both simultaneously.