Glycolysis literally means the "breaking down of glucose". This can occur without oxygen, meaning it's anaerobic. This occurs in the cytoplasm of the cell, which is the main open part of the cell.
No Oxygen?
This can occur without oxygen. If there is no oxygen, they body starts undergoing lactic acid fermentation. Lactic acid fermentation is similar to glycolysis, but the pyruvate turns into lactic acid at the end.
Oxygen?
If there is oxygen present, it goes onto the Krebs Cycle then to the Electron Transport Chain
Process
There are two phases in glycolysis, the so-called "Investment Phase", and the "Payoff Phase"
Investment Phase
In the investment phase, 2 ATPs are used. In this phase, glucose is turned into 2 three-carbon molecules (each with a phosphate group), known as GADP or PGAL. The scientific name is Phosphoglyceraldehyde, or Glyceraldehyde-3-phosphate.
Payoff Phase
In the payoff phase, the GADPs turn into pyruvates, or pyruvic acid. Each GADP conversion produces 2 ATPs and reduces an NAD+ to an NADH. So, in total, they produce 4 ATPs, and two NADHs. NADH is used later to donate electrons, so it is a valuable part of the electron transport chain, the last part of cellular respiration.
Net Result
At the end, glycolysis produces 2 net ATP (4 total) , and 2 NADHs.