Strength measures the muscle and physical power of your character. This ability is especially important for fighters, barbarians, paladins, rangers, and monks because it helps them prevail in combat.
Dexterity measures agility, reflexes, and balance. This ability is most important for rogues and warlocks, but also for characters who typically wear light or medium armor (barbarians and rangers) or none at all (monks, wizards, and sorcerers), or for any character who wants to be a skilled archer.
Constitution represents the health and stamina of your character. High constitution increases the number of hit points a character has (affecting how much damage they can take), and this makes it important for everyone, but especially so for fighters. If constitution ever increases, hit points increase retroactively as well. Spellcasters also need strong constitution to keep their spells from being interrupted during combat.
Intelligence determines how well your character learns and reasons. Intelligence is important for wizards because it affects how many spells they can cast, how hard their spells are to resist, and how powerful their spells can be. It is also important for any character who wants to have a strong assortment of skills, however increasing your Intelligence will not grant bonus skill points retroactively. Keep in mind that sorcerers, bards, and warlocks do not use Intelligence for any of their spellcasting - Charisma is used instead.
Wisdom describes a character's willpower, common sense, perception, and intuition, whereas Intelligence represents the ability to analyze information. An "absent-minded professor" has low Wisdom and high Intelligence. A simpleton with low Intelligence might nevertheless be very perceptive (have high Wisdom). Wisdom is important for clerics and druids, affecting the strength and number of their spells, and is also significant for paladins and rangers. It is very important for monks as well, modifying many of their abilities. The wise character is capable of intuitively understanding others, occasionally having insights into their motivations.
Charisma measures a character's force of personality, persuasiveness, ability to lead, and physical attractiveness. It represents actual personal strength, not merely how one is perceived by others in a social setting. Charisma is most important for warlocks, paladins, sorcerers, and bards since it determines what spells they can cast, as well as the effectiveness of those spells. It is also important for clerics, as it affects their ability to turn undead. All characters benefit from having a high charisma when speaking with others in the world since it modifies the player's Diplomacy, Bluff, and Intimidate skill checks. Keep in mind that wizards use Intelligence for their spellcasting, not Charisma.