Look at the beginning of the Wikipedia page for "Acid," and you'll notice the extremely non-helpful definition:
"An acid (from the Latin acidus/acēre meaning sour) is a substance which reacts with a base."
Ok...so that's great. But what is an acid really? To answer that question, we have to look at the structure of water. Everyone knows the chemical formula for water--H2O. Of course, that means that one water molecule has two hydrogen atoms bonded with one oxygen. However, those bonds can be broken, and the atoms can shift locations. Water molecules can exchange protons with each other because the oxygen has a higher electronegativity than the hydrogen atoms.When this exchange occurs, then one water molecule becomes positively charged (H3O+) and the other negative (OH-).
What does this have to do with acids? Well, when the compound HCl is added to water, the same process occurs. HCl separates into a hydrogen ion (H+) and a chlorine ion (Cl-). These hydrogen ions create the positively charge water molecules (H3O+). This in turn causes the solution to become more acidic because the ratio of H3O+ to OH- is shifted in favor of the hydronium ions (H3O+). A base, on the other hand, does the opposite. It doesn't add protons to the system; instead, it takes protons away and shifts the balance in favor of the hydroxide ions.
Acids have a reputation for being bad--after all, they can burn through your skin, right? However, some gentle acids include soda and rainwater, going all the way down the pH scale to not-so-gentle acids like stomach acid. Bases, on the other hand, can be just as severe in reaction--they include ammonia and Drano.
Acididy and basicity are measured on the pH scale. The pH scale corresponds to the concentration of hydronium ions in a solution. It's actually just the absolute value log of the concentration of hydronium ions in a solution. For example, in pure water, the concentration of hydronium ions is 1x10^-7. Since the absolute value of the log of 1x10^-7 is 7, the pH of water is 7.
Titration is a technique that allows us to determine the concentration of an unknown reagent by using a known concentration of another reagent that reacts with the other reagent. Titration can be used to determine of unknown acids or bases. It's called titration because the reagent with the known concentration is called the titrant. (That comes from a Latin word, but let's not go into that.) The two solutions are allowed to react, and once the reaction has completed (the "equivalence point" has been reached), all of the titrant added to the unknown solution and the unknown solution have reacted. Here, we can find that:
(Normality of titrant)(Volume of titrant needed) = (Normality of unknown)(Volume of unknown)
(Normality is a measure of a solution's ability to release ions into another solution. For example, in an acid reaction, the normality of H2SO4--sulfuric acid--is 2, because there are 2 moles of H+ per mole of sulfuric acid.)
Now, because the titrant is a known solution, we know its normality, and we can measure how much of it was needed to finish the reaction. We also can measure the volume of the unknown. This leaves us with one unknown--the normality of the unknown reagent. Because there is only one unknown in this equation, we can easily solve for it and determine the concentration of the reagent.