Hydrophobic means literally 'fear of water'. This isn't just referring to that rather unusual state that can occur in some diseases (e.g. rabies) where you become fearful of water. Rather, it's talking about a tendency of chemicals to mix badly with water - in fact, it tends to refer to those chemicals that mix well with fats. Why?
The basic rule of thumb is that similar things mix well together. You might know, for instance, that oil mixes well with oil and water mixes well with water, but oil and water don't mix - oil floats on the top. This is because oil is hydrophobic, whereas water is hydrophilic. The reason that these substances work like this is because of the way their molecules interact.
Take, for instance, a normal biological membrane. It's made up of lots of phospholipids - a phosphate group with chains of lipids or fats shooting off the end. The fats are long hydrocarbon chains - that is, long chains of atoms with carbon running along the middle, and each carbon atom connected to two hydrogen atoms and the adjacent carbons. There is very little variability. In fact, there is so little variability, that there is no significant polarity in the fats at all.
Water, on the other hand, has an oxygen atom and two hydrogen atoms. Due to the electonegativity of the oxygen atom, there's a slight difference in the charge from one end of the molecule to the other. As explained in the section on polarity, the water molecule is polar.
Because the water molecule is polar but there is little difference in the fat molecules of a phospholipid, they don't mix well. The polar molecule wants to mix with things which have a slight charge or some kind of difference from one end of the molecule to the other. However, the fat molecule wants to mix with things which are neutral.
This is important in a whole load of ways. For instance, in biological membranes, the hydrophobic phospholipid will allow things through if they are hydrophobic (e.g. steroid hormones) because the fats in the middle of the membrane will be happy to mix with other hydrophobic things. However, other substances which are big and polar can't squeeze through because they don't mix with the fats. In biology, some of the most important hydrophobic chemicals are fats and oils.
Hydrophilic, the opposite of hydrophobic, means 'water-loving'. Again, this isn't just a normal human being who loves water. It's the tendency of chemicals to mix well with water. Those substances which are polar mix well with other polar things. Because water is so important, everything is described in relation to water - and so polar molecules are described as hydrophilic.
Something is amphipathic if it is both hydrophobic and hydrophilic. This is possible when it contains parts of molecules that are hydrophobic and parts that are hydrophilic, and it means that the substance can dissolve both in water and in something like oil.