Ruy Lopez

This is one of the oldest openings known and it is still widely used, since it gives the white player a lasting initiative at the outset of play. Recently, it has fallen in esteem to the Sicilian Opening, which we will discuss next, since black tried to avoid the Ruy Lopez, and prefers an opening in which he doesn't have to cede the initiative immediately. In chess, remember, it takes two to tango!
The standard opening moves are:
1. e4 (P-K4)          e5 (P-K4)
2. Nf3 (N-KB3)      Nc6 (N-QB3)
3. Bb5 (B-N5)

The most common variation of this opening is the Morphy Defense, which answers with 3...a6 (P-QR3).

Sicilian Defense

This defense is considered best these days for black, for it gives the black forces immediate counter play, and isn't as passive as the Ruy Lopez for the defender.
The standard opening moves are:
1. e4 (P-K4)          c5 (P-QB4)
2. Nf3 (N-KB3)      d6 (P-Q4)
3. d4 (P-Q4)          cxd4 (PxP)
4. Nxd4 (NxQP)    Nf6 (N-KB3)

The above moves are common to the Najdorf and Dragon variations of the Sicilian. There are other variations known as the Taimonov, Paulsen and Scheveningen. We suggest that you study Modern Chess Openings as you become more involved with chess.

French Defense

Unlike the Sicilian, this is a quieter opening in which black tries to close up the position. It is best suited for non-aggressive players who prefer a solid, rather than an attacking defense. 
The standard opening moves are:
1. e4 (P-K4)          e6 (P-K3)
2. d4 (P-Q4)          d5 (P-Q4)
3. Nc3 (N-QB3)     Nf6 (N-KB3)

There are two variations of the French that are popular - the classical and the Winawer.

Queen's Gambit Declined

Although a pawn is sacrificed at the beginning in this opening, which gives it the term gambit, the sacrifice is temporary, for white easily regains the pawn. This opening is used extensively by modern masters. 
The standard opening moves are:
1. d4 (P-Q4)          d5 (P-Q4)
2. c4 (P-QB4)        e6 (P-K3)
3. Nc3 (N-QB3)     Nf6 (N-KB3)
4. Bg5 (B-KN5)     

Variations of this opening are the Orthodox, Exchange, Tarrasch and Ragozin. Also the Slav and Semi-Slav. Another variation is the Queen's Gambit Accepted, where, in Black's second move, it takes the pawn at c4 with its d pawn. Again, refer to Modern Chess Openings.

Queen's Indian Defense

The purpose of this opening is to control the center, particularly the square e4, with both black's queen's bishop and king's knight. To do this, black financhettos on his queen's side. A financhetto is the movement of the bishop to the b7 square, where it controls the long diagonal leading to the white king's position.
The standard opening moves are:
1. d4 (P-Q4)          Nf6 (N-KB3)
2. c4 (P-QB4)        e6 (P-K3)
3. Nf3 (N-KB3)      b6 (P-QN3)
4. g3 (P-KN3)        Bb7 (-N2)
5. Bg2 (B-N2)        Be7 (B-K2)
6. 0-0

There are analogous systems based on the Indian defense, known as the Nimzon_Indian and King's Indian.

Some other modern Queen's openings are the Modern Benoni and the Grunfeld Defense. What is important is to know some of the more popular openings used today, and then to go into further study of the variations. Combining this with the opening principles will make you into a competent player rather quickly.