Overview
The Relative Strength Index ("RSI") is a popular oscillator.  It was first introduced by Welles Wilder in an article in Commodities (now known as Futures) Magazine in June, 1978.

The name "Relative Strength Index" is slightly misleading as the Relative Strength Index does not compare the relative strength of two securities, but rather the internal strength of a single security.  A more appropriate name might be "Internal Strength Index." 

Interpretation
When Wilder introduced the Relative Strength Index, he recommended using a 14-day Relative Strength Index.  Since then, the 9-day and 25-day Relative Strength Indexs have also gained popularity.  The fewer days used to calculate the Relative Strength Index, the more volatile the indicator.

The Relative Strength Index is a price-following oscillator that ranges between 0 and 100.  A popular method of analyzing the Relative Strength Index is to look for a divergence  in which the security is making a new high, but the Relative Strength Index is failing to surpass its previous high.  This divergence is an indication of an impending reversal.  When the Relative Strength Index then turns down and falls below its most recent trough, it is said to have completed a "failure swing."  The failure swing is considered a confirmation of the impending reversal.

In Mr. Wilder's book, he discusses five uses of the Relative Strength Index:

Tops and Bottoms.  The Relative Strength Index usually tops above 70 and bottoms below 30.  It usually forms these tops and bottoms before the underlying price chart.
Chart Formations.  The Relative Strength Index often forms chart patterns such as head and shoulders or triangles  that may or may not be visible on the price chart.
Failure Swings (also known as support or resistance penetrations or breakouts).  This is where the Relative Strength Index surpasses a previous high (peak) or falls below a recent low (trough).
Support and Resistance.  The Relative Strength Index shows, sometimes more clearly than price themselves, levels of support and resistance.
Divergences.  As discussed above, divergences occur when the price makes a new high (or low) that is not confirmed by a new high (or low) in the Relative Strength Index.  Prices usually correct and move in the direction of the Relative Strength Index.