My mother has a box type air conditioner which I tried to replace.
There is a box that is made of metal that sticks out from the house
which I mounted the new air conditioner in. However, she thinks the new
unit has intakes on the side which will hit up against the box it is
enclosed in. Are there special box type units that vent in the back ?
She was told those are older style units and that central A/C or window
style is more common.
The water generated from condensation is also an issue.

You may be talking about a through wall unit v window unit.
As I understand it -
Through wall units are arranged to vent front & back while window units
vent to the side.
Newer equipment is arranged to deal with condensate without dripping
from the unit.
There have been lots of threads about this. Search around the NG.
TB

.. Are there special box type units that vent in the back ?
She was told those are older style units and that central A/C or window
style is more common.
The water generated from condensation is also an issue.

Yes,there are A/C units specifically for through-the wall installation.
Normally if you have the sleeve in the wall already, just get the name
of the sleeve or the original A/C that was in it ( usually Friedrich
and Fedders are common) and the local appliance stores should have
them. The wall A/C's vent through the back. If you try to operate a
window A/C in a sleeve, it will not run correctly.
The newer A/C's made today are designed to hold the condensate in the
pan to allow the coil to cool more efficiently, so the condensation
should be no problem.

My mother had gone to sears and she said they didn't have what she
needed. I went to home depot and explained I had a box type and the
guy gave me a unit, and I had very little time to install it. I just
plunked it in. later my mother looked at the instructions and saw it
vents to the side. The unit leaves an extra 5 or 6 inches as it is not
as wide as the box. Either that would be sufficient or I may cut out a
hole in the box sides to vent it, and figure out a way to insulate
around it farther forward or back from the side vents for winter.
Either that or I would return it, but what stores might have through
wall units ?