Hydrophobicity literally means "fear of water". The opposite of this is hydrophilicity which means "love of water". These two terms denote the properties of molecules or parts of molecules to bond with water (hydrophilicity) or reject water (hydrophobicity). This effect is seen when one puts a drop of oil on water. The oil (hydrophobic) remains on the surface and sticks together instead of mixing with the water (hydrophilic). In cells these properties are essential. Most of the cell is hydrophilic but there are hydrophobic borders (membranes) which compartmentalize the cell into different spaces and hereby separate reactions. Of course there must be some sort of interaction between compartments and this is facilitated by membrane proteins (look here to read up on membrane proteins).
These membranes are formed with molecules with hydrophobic properties, such as phospholipids, cholesterol and membrane proteins. The phospholipids and membrane proteins also have hydrophilic properties which stand in contact with the cytosol or the extracellular matrix.