The concepts of electrophilicity and nucleophilicity have to do with the formation of bonds in chemical reactions. In organic chemistry, most bond-forming reactions can be classified as homolytic or heterolytic. For heterolytic reacctions, one partner receives two electrons from another partner simultaneously to form a bond between them. The electron pair donor in a heterolytic reaction is the nucleophile and the electron pair acceptor is the electrophile. Specific atoms on the reacting partners are sometimes called nucleophiles/electrophiles or nucleophilic/electrophilic centers.
So when confronted with a reaction, how do you predict which one is the electrophile or the nucleophile? If you know what's donate electrons to what, then you have the answer, from the definition above.
If, however, you don't know beforehand, you can often tell which one will donate electrons to another. Nucleophiles tend to have several (two or more) lone pairs of electrons (oxygen, for instance, often has two; nitrogen often has one). Also, nucleophiles tend to be electronegative elements, which can give them a partial negative formal charge (by polarizing bonds and pulling electrons towards them). If a second-row or third-row nonmetal atom (such as nitrogen, oxygen, sulfur, phosphorous, and so on) has a full octet and some lone pairs, these atoms are often nucleophilic.
Electrophiles are often electron-poor, without a full octet. Usually, they are less electronegative or have many electronegative atoms bonded to them, giving them a partially positive formal charge. Sometimes, if some part of the molecule has a large anti-bond, then that part of the molecule can be electrophilic.
Some examples are in order. How about in the following reaction?
On the left hand side, ammonia (NH3) has a lone pair of electrons that aren't bonded to anything, and the carbon on the methyl iodide has no electron pairs free. On the other hand, the carbon-iodine anti-bond is very large on the carbon side, and so it is likely that the carbon is an electrophilic center, while the nitrogen on the ammonia is a nucleophilic center. This means that the ammonia is a nucleophile while the methyl iodide is an electrophile. And we see that the final compound has a bond between carbon and nitrogen, which confirms our predictions.
Let us try another example. The following example has an acid-catalyzed geminal diol formation from acetone (dimethyl ketone) and water.

In this case, the water has an oxygen with two lone pairs of electrons, but the acetone oxygen also has two lone pairs, so this does not tell us right away which reactant is the nucleophile. Here, however, the carbonyl carbon is partially positively charged, because it is doubly bonded to oxygen, which is a more electronegative element. Meanwhile, the oxygen on the water molecule is more electronegative than the hydrogens, and so has a partial negative charge. This gives us that the nucleophile is the water's oxygen and the electrophile is the carbonyl's carbon.
If you already know how the reaction will proceed, as we do here, then the answer can come much more easily. The main bond formed here is the carbon-oxygen bond, and since the carbon that eventually forms that fond has no free lone pairs and is partially positively charged (because it is bonded to the oxygen), it must be an electrophile, and since the oxygen that forms that bond has two free lone pairs as well as a partially negative charge, it must be a nucleophile.
To finish up, most of the time, the easiest way to tell a nucleophile from an electrophile is to look at partial positive and negative formal charges created by polar covalent bonds. The negatively charged ones are nucleophiles, and the positively charged ones are electrophiles. 