Open-drain: the chip can only pull a line low; it relies on an external pull-up resistor to make it high. Push-pull: the chip can drive both states actively. I2C is open-drain, SPI is push-pull.
How it looks
Drawing
Left: one switch plus a resistor. Right: two switches.
How it works
Open-drain can only pull low; an outside pull-up makes the high. Slower, but many chips can share one wire safely. I2C works this way.
Push-pull drives high and low itself. Fast, but two outputs on one wire can fight (contention). SPI works this way.