Assuming the resource is located on a tile inside the culture borders of one of your cities, send a worker to that tile. When he arrives, the relevant tile improvement option will be highlighted and you can instruct the worker to improve the tile (e.g., build a plantation, dig a mine, etc.). That will provide that resource to your empire. There are 3 different types of resources in the game: luxury resources, strategic resources, and bonus resources.
Luxury resources main benefit is that they provide 4 happiness per unique luxury (extra copies are only useful for trading purposes -- see more below), although in some cases the existence of an improved luxury in a city will enable you to build a special building in that city (e.g., gold or silver allows you build a mint, ivory allows you to build a circus).
Strategic resources allow you to build units and buildings that require those resources (e.g., 1 horse allows you to build 1 horse unit, 1 iron allows you to build 1 frigate, 1 coal is required for 1 factory, etc.). Therefore, extra copies of strategics are always useful, since you will probably want to build more than 1 horse unit, or factory, or whatever, in your empire.
Luxury and strategic resources can be traded with other civs, which means you can get gold from other civs or trade for a resource that you need (e.g., you have two copies of incense and an AI civ has two copies of wine -- trade benefits both of you). Bonus resources (like fish, cattle, sheep, wheat) cannot be traded, but do enhance the yield of the tile on which they are located.
Trade is a separate system from the system that relates to "working" improved tiles. In every city, the city center tile is automatically "worked" by a "phantom" citizen and each other citizen can be assigned to "work" one tile (or "work" one "specialist slot", after you build buildings with specialist slots). Each city's yield of food, production, science, culture, etc. is the sum of the yields of tiles that are "worked" by that city's citizens, plus any yield from buildings or wonders and any yield multipliers or bonuses from social policies, technologies, buildings or wonders. As you have noticed, when a tile has an improved luxury, strategic resource or bonus resource, the improvement enhances the yield of that tile. Note that you get that enhanced yield even if you have traded away that resource.
The city view screen allows you to manage what tiles your citizens are working. You can either automate citizen assignment, by setting the "city governor" to one of the standardized focuses (default, food, production, science, etc.), or you can manage your citizen assignment manually, by checking the citizen management box in the top right of the city view screen. What you see there will be very illuminating.
