To expand is to build a new command center, nexus, or hatchery at an unoccupied base. The timing of when and where to expand is extremely important, and is one of the 3 main strategic choices any player has (alongside teching and attacking). Expanding is a risky move - it costs 300-400 minerals, exposing new workers as well, and stretches a player's defenses to cover more territory. However, it's also a necessary move. In standard StarCraft play, bases run out of minerals, so a player that doesn't expand will quickly run out materials to tech or attack and be overrun. Because of this, in long games it's very common to see 6-7 expansions per player, depending on the racial matchup.
Strategically, pro coaches encourage their players to only expand to locations they can defend, while newer players can usually sneak a new base in without the other player realizing. It's also important to know exactly when and where your opponent has expanded, so that you can time your attacks to take advantage and predict their upcoming moves.