What happens if I connect two lead acid batteries in series, one fully charged and one half charged?
Short-term, this isn't a problem. You'll end up with ~24V powering the load.  
Long-term, this is a bad idea. Once the half-charged battery gets drained, the full-charged battery will now be at half charge and will start running the drained battery in the reverse direction. Lead-acid batteries hate to be deep-discharged. The lead plates will corrode and you'll lose capacity on them permanently if not destroy the battery entirely.


To prevent the second battery from running backwards or even being deep-discharged, make sure you balance the batteries before connecting them in series and running them to the load. A poor man's semi-decent attempt to balance them would be to just put both batteries in parallel for a long time to transfer some charge from the full battery to the half-charged one.

Another option is to always monitor the half-charged battery to make sure that you never deep discharge it. You'll have problems charging it as well though. Upon charging, you'll likely end up over-charging the full battery which is just as destructive, but in a different physical mechanism.