Basics: lanes, positions and roles

How the map works and who does what in a team of five.

The map and the goal

Two teams of five heroes — the Radiant and the Dire. The bases sit in opposite corners and are linked by three lanes: top, middle and bottom. Creep waves walk the lanes, guarded by towers. To win, break through the towers and barracks and destroy the Ancient in the enemy base.
Between the lanes lie the jungle with neutral creeps and the river with runes. The safe lane is bottom for the Radiant and top for the Dire; the offlane is the opposite one.

Position 1 — carry. Plays the safe lane, takes the most gold and becomes the team's main strength in the late game.
Position 2 — mid. Alone in the middle lane, gets levels early and is the first to influence the map: ganks, runes, early fights.
Position 3 — offlaner. Survives the hard lane against the enemy carry, disrupts their farm, later initiates fights and soaks damage.
Position 4 — soft support. Roams, helps lanes, places wards; often the one with Blink Dagger and initiation items.
Position 5 — hard support. Gives gold to the team, buys wards and save items, protects the carry in lane.

A position is a place in the team; a role is what a hero is good at. Valve and statistics sites describe heroes by roles: carry (scales with items), support, nuker, disabler, initiator, durable, escape, pusher. A good lineup covers every important role and does not stack the same weakness.
Each hero's roles are on their knowledge base page, and the Team builder shows which roles a game plan needs and which are missing.
Team builder

Gold comes from last hits, kills, destroyed buildings and bounty runes; experience comes from creeps and heroes dying near you and from wisdom runes. A deny is a last hit on your own creep, so the enemy gets no gold and less experience.
A team's economy is about splitting the farm: the carry and mid take more, supports less, and in return provide vision and safety.