What are some antenna designs that balance manageable size/shape with effectiveness for use in an apartment?
The answer depends highly on what you wish to accomplish. What sort of frequencies, how high power, distance you're going after, etc.
VHF/UHF (145 MHz and up) antennas are small - if your window points to the right direction, and you live on the 6th floor, a small yagi in front of it can be good for serious DX when the propagation is right. You can also hang a dipole in front of the window, which will be good for local contacts and working the repeaters. Copper tape isn't expensive, you can use it to tape a dipole or J pole directly on the window surface, and solder a coaxial on the feed point.
HF antennas will never work very well inside, but even a bad antenna is always better than no antenna at all. High power is out of the question - some of the power will end up on the building's power and phone/network cables, and easily cause RF interference.
Magnetic loops are an interesting option for HF work - they are relatively narrow-band and tunable, and have other properties which make them more immune against electrical noise from household appliances, relays, and electronics. They are relatively small and can be placed in front of a window or wall. They need to be constructed out of thick metal tube or thick hardline coaxial cable, which makes them a bit heavy, but they still don't take more space than a heavy hula hoop. To run more than a few watts you'll need a high-voltage tunable capacitor, but QRP is fun to work too. With WSPR and similar digital modes for weak signals you can quite easily get proper DX contacts.
Dipoles work too, but they'll pick up more static / electrical noise than loops. They're also pretty long for lower frequencies.
For higher frequencies like 2m, two pieces of wire soldered to a BNC connector and hung by your window make a great dipole!
There are also various "apartment window flagpole" antennas that can do the job.
For DX, you run into a problem with grounding... there's no way to make a counterpoise or ground radials big enough. Fortunately, neither dipoles nor magnetic loops need a counterpoise or ground plane.
G4FON's 20/30/40m band magnetic loop for travelers looks nice: http://www.g4fon.net/MagLoop.htm
If that's too complicated, Scott Harden's DIY Stealth Apartment Antenna may also be something worth trying... it's essentially a 20/40m dipole folded to fit an apartment: http://www.swharden.com/blog/2010-02-07-simple-diy-stealth-apartment-antenna-for-20m-and-40m/