Either AC or DC should work, but be careful with either. For control with Arduino or other microcontrollers, the first consideration is isolation. The gate will likely require current and voltage, far outside of an Arduino's capacity.
Perhaps the easiest solution would be to use a relay connected through an opto-isolator (Arduino->opto-isolator->relay->motor) to drive a simple on/off motor. Many other possibilities exists, including motor-dirvers/shields, and servos. However, they are more complex than appears to be needed. Unless you need to control the speed and/or angle of the gate, a simple relay should do the trick.
To pick a motor, consider the weight that needs to be moved, and the power sources available. A balsa wood gate would require very little in the way of power, but would break easily, perhaps not being much of a barrier. A steel gate would be an effective barrier, but would require a far more robust motor and power supply. I'd go as light as possible. This will be cheaper, and easier (less dangerous) to work with.
Plenty of other considerations will affect the choice as well. Does the gate need to stay up if power is cut, or should it return to a closed position ? Does it need to sense an object below when it is closing ?
You may also consider salvaging a motor from an old drill. This would be more than powerful enough for most gates, and you would only have to bypass the switch. Indeed, it may be more powerful than you want.
Finally, consider counter-weighting the gate (hinge it a bit offset from the end, and add weight to the short end if needed). This will reduce the amount of power needed, but will increase the size a bit.