There are several ways to define acids, but the simplest classification describes an acid as a substance that donates hydrogen ions to water. The higher the concentration of hydrogen ions in water, the more acidic it becomes.
In pure water, water molecules will sometimes spontaneously dissociate to form a hydrogen ion (H+) and a hydroxide ion (OH-). This process is called autoprotolysis. Hydroxide ions, however, can also take a hydrogen ion from another water molecule to reform H2O. Both processes are ongoing and in pure water are at an equilibrium such that there are a mere 1x10 to the -7 hydrogen and hydroxide ions for every liter of water.
Since pH is defined as the negative log of the hydrogen ion concentration, the pH of pure water is 7 or neutral. When an acid like sulfuric acid (H2SO4) is added, however, the acid donates hydrogen ions to water molecules, increasing the concentration of hydrogen ions and thus decreasing the pH.
Water with a low pH has a high concentration of hydrogen ions. The hydrogen ions aren't merely floating around untethered, however; each hydrogen ion combines with a water molecule to form a hydronium ion, H3O+ . Hydronium ions act as acids by donating hydrogen ions to other substances, so they are what makes the water acidic.