Electrical – Outlets not working but have power

As I said in the comments to the OP and to Aaron's answer, there are several wiring failures that could cause an NCVT (non-contact voltage tester; usually a pen-looking thing that beeps when you point it at a live electrical wire) to light up, and some that even a plug tester can miss, but that would prevent a "real" load from being plugged in. The first thing to do is get a 3-lamp outlet tester (preferably with GFCI testing) to supplement your NCVT. The tester will correctly diagnose most of these things:

• Open Hot: the black wire is present in the box but disconnected from the plug (or otherwise not making good contact). An NCVT can still light up if the wire is close to the faceplate. However, a plug tester will diagnose the issue. The fix is to flip this outlet's breaker, open up the plate and make sure all wires are firmly connected to the proper terminals on the outlet.
• Open Neutral: the white wire is present but not properly connected to the outlet. Again, a NCVT will light up if the hot is properly connected (maybe even if it isn't), but a plug tester will properly diagnose the issue. The fix is the same as for an open hot.
• Broken/Worn/Bent Outlet Contacts: The wires are hooked up correctly, but the outlet's contacts have poor or no continuity with the plug's blades. You may hear arcing (a rough crackle) as you plug in the tester or an electrical device. A tester MAY show correct wiring at a glance, but if you jiggle the tester the lights will blink off. Replacing the outlet is the solution here.
• Neutral-Ground Swap and High-Resistance Ground: This would tend to affect an entire house's wiring, but it can be localized as well. What is happening is that the white and bare wires have been switched, and in addition, the ground (which normally ties back to the bus strip in the service panel and has full continuity with the neutral) has lost some continuity back to the service panel. There are several situations in which this can happen, most of them involving attaching grounds to house plumbing and then changing that plumbing along the way. An outlet tester may not diagnose this; the expected read is an "open ground" or "open neutral" (remember this would mainly be a problem if the neutral and ground are switched), but if there is some continuity then the tester may not show a problem.
• Poor Continuity Of Hot Or Neutral With Outlet: One or more wires is making very poor contact with the outlet terminal. A plug tester may miss this; there is enough continuity for the tester (which only draws a few milliamps for its LEDs) but even a clock radio would have issues with the load induced by the poor continuity. The solution is to rewire the outlet securely, and if the wiring is sound on the plug then to check other wires in the box (with the breaker off of course). The next most likely culprit is the "bundle" of neutral leads that's normally in an "upstream" J-box; make sure all wires are wound tight and properly secured with a wire nut and electrical tape.

If you are using a simple outlet tester you could get a false positive. You should always use something that draws some decent current, like a lamp. Here is a related story. I unplug the electric clothes dry recently while cleaning the ducting. My wife plugged it back in afterward. From then on, although the drum turned fine, the heat was intermittent. After checking a couple other things, I pushed the plug all the way in. Apparently, the motor ran on 120V, where the heater coil ran on 220V. Needless to say, I checked the plug and outlet for burns.