When a windmill is producing charge power its alternator/generator/etc. causes the windmill to slow down. If the attached batteries become full and the charge controller stops taking power from the windmill then that braking force will no longer be present and the windmill will spin much more quickly and potentially destroy itself. Thus a charge controller designed for windmills will employ a dummy load to keep that braking force present and prevent those overspeed conditions.
For the wind turbine you have linked (which I would not
recommend) you would just need a diode to prevent the battery from discharging into the turbine. The windmill would spin freely below the battery + diode Vf and then slow down when it reaches the RPM that creates the voltage high enough to charge the battery.
The picture of the battery you have linked shows that its "standby" voltage is 13.5 to 13.8V. When trickle charging you would aim for 13.5V in hotter climates and 13.8V in colder climates.
You asked if the Ah rating of a battery affects how it should be charged -- yes it does. The general rule for all batteries is that, for the best lifetime, they should be charged at a 10 hour rate; e.g. a 10Ah battery would be charged at 1A for 10 hours.
Jose demonstrates a basic charge controller in the summary in this thread: http://forum.arduino.cc/index.php?topic=254591.0
Your best bet is going to be to use solar panels; windmills, at a small scale, are not practical. If you can provide more details on how much power your project is drawing then you'll get better answers as to what to do.