The rule of thumb is, a standard 15-amp circuit should have no more than 12 things plugged into it. That's 12 light bulbs (some count fixtures) and plug outlets. This is very conservative, but ensures you won't throw a breaker in any normal circumstance.
For a more exact specification, a 15-amp circuit at the U.S.-standard 120V will provide 1800W of power (the equation is very simple; V * A = W, so 120VAC x 15A = 1800W). You can normally exceed 1800W for "transient" periods (peaks, less than 1sec above the limit), as a non-GFCI/AFCI breaker is usually "slow-trip". So, to determine your wattage requirements, simply add up the wattage draws of all the things you'll be plugging into that circuit.
You see how the "12 things" rule can be very conservative for certain types of circuits: 12 60W incandescent light bulbs is only 720W, less than half the max output. Even 100W bulbs would only total up to 1200W. And if you've been a green homeowner ans swapped out as many incandescents as possible for CFLs, those 100W bulbs are actually only drawing 26W each, for a draw of a whopping 312W.
However, a circuit almost never has just light bulbs. Consider the circuit into which you plug your home theater. The average plasma TV draws about 300W, with some 60" monsters drawing up to 600W. Then add a 300W home theater audio system, Blu-Ray player (200W), gaming consoles (150-200W each), DVR (50W), desktop computer (depends on the tower but a decent gaming rig will have at least 500W) and even a modest home theater could be drawing over half of the available power of the circuit from a single 2-plug wall outlet. THEN, add light fixtures that run on the same circuit (still perfectly legal in many situations, though the kitchen and bedrooms should have dedicated circuits for outlets and/or "built-in" appliances), and when you plug that 9-amp vacuum cleaner (1080W JUST FOR THATjust for that) into the next outlet, turn it on and trip the breaker, you should no longer be wondering why.