Glycolysis is the breaking down (lysis) of glucose (glyco) into pyruvate (also known as pyruvic acid).

It is the first part of cellular respiration, where cells use glucose as fuel for energy (making ATP).
Glycolysis takes place in the cytoplasm of the cell, and some energy is made in the process. This part of respiration does not need oxygen, so it is anaerobic.

Glycolysis converts glucose (a 6-carbon molecule) into pyruvate (a 3-carbon molecule) in a few steps.

Step 1

Glucose is first phosphorylated (phosphate group, P, added) by a kinase enzyme, to Glucose-6-phosphate, using 1 ATP molecule. (Read about ATP and ADP)

Step 2

Glucose-6-phosphate is then rearranged (by an isomerase enzyme) into fructose-6-phosphate.

Step 3

Another ATP molecule is used to add a second P to fructose-6-phosphate, making fructose-1,6-bisphosphate.

Step 4

Fructose-1,6-bisphosphate (6 carbons) is not stable, so it divides into 2 molecules : glyceraldehyde 3-phosphate (3 carbons)

Now comes the "pay-off phase" where ATP is actually made, instead of consumed!

Step 5

Each 3-carbon molecule undergoes phosphorylation (this time with inorganic phosphate) and a hydrogen is removed from each molecule. An oxygen carrier, NAD+ takes this hydrogen and forms NADH + H+. (NADH produced here goes to the electron transfer chain of cellular respiration). 2 NADHs are produced per glucose (since 1 glucose --> 2 glyceraldehyde 3-phosphates).

Step 6

The P that was just added is enzymatically transferred to ADP, forming ATP. This step is called substrate-level phosphorylation. ADP is required for this step to happen! 2 ATPs are produced per glucose molecule.

Step 7

Water is also removed from each 3-carbon molecule.

Step 8

A final step converts the 3-carbon molecule into pyruvate, by transferring the last phosphate group to ADP, forming another ATP.

So in summary:

2 ATPs are first used to "activate" glucose.

In the pay-off phase:

4 ATPs
2 H2Os
and 2 NADH + H+s

are made.

Therefore, on the whole:
glycolysis forms 2 ATPs, 2 water molecules and 2 NADH+ H+s (which makes another 4 ATPs in the electron transfer chain).

If you want to visualise these steps better, check out some of these animations!
1. A simple animation by Graham Kent of Smith College
2. Another good animation, with short quiz from Anatomy & Physiology, 7e, by Seeley, Stephans & Tate