The answer to your question depends on the kind of cell we are talking about. Autotrophic cells such as plants and certain bacteria usually generate ATP through photosynthesis or chemosynthesis. They basically take energy from a non-organic source, such as the sun or various inorganic chemicals, and convert it to energy that is "usable" by the rest of the cell and other organisms. Ultimately this non-organic energy is used to drive what is called an "electron transport chain." Many enzymes are "embedded" along the electron transport chain. These enzymes, called ATP synthase, add a phosphate to ADP to make it ATP. If you're wondering where the ADP comes from, it is released as a by-product in chemical reactions in the body that require ATP. So there is a cycle of the following form:

Plants can also undergo cellular respiration to produce ATP, but generally, heterotrophic organisms, which obtain their energy by "eating" other organisms, generate ATP during cellular respiration. Cellular respiration converts macromolecules, such as glucose or lipids, to smaller molecules and energy that is "usable" by the cell. The cellular respiration pathway has its own version of the electron transport chain -- with different "electron carriers" and ATP synthase enzymes than those found in photosynthesis.

Cells have ion channels that generate ATP - a proton moves through a gradient through the protein in the membrane, and this causes a confirmational change elsewhere in the protein that turns ADP into ATP.