Generally two kinds:
1. High gain, possibly with schmitt trigger. Square wave output, standard logic output pin. Needs a series termination resistor to drive the crystal and limit drive power (typically 1.5kΩ or so). This requires additional parts, but can be used with pretty much anything you can hook onto it: crystals, ceramic resonators, even LC resonators.
2. Limited gain, usually a single stage inverter. No series resistor is needed. Gain depends on size of the inverter, which can be too low for some crystals. ST MCUs for example tend to be on the small side, so need a higher ESR crystal (and smaller loading capacitors) than others do. The drive strength may also be selectable from software (e.g. TI MSP430s often do), or from hardware (e.g., ATMEGA fuse settings).
The signal level is small (100s mV?), limited by the transfer function of the inverter: gain is highest for small signals, and drops at larger signal levels. This also causes odd mode distortion, which is why the waveform isn't quite sinusoidal, but a bit squashed.
Tim