Rear-mount.
These child bike seats come with a rack.
The rack is bolted to the back of your bike.
Important:
Your bike must have threaded mounting points for a rack. Most (but not all) aluminium frames will have these. Most carbon frames will not - but it's worth checking.
The child seat sits on top of the rack.
And because there's extra support, these child seats can handle more weight (up to 22kg for the Topeak Babyseat II for example) than front-mount child seats.
The rack stays on the bike and it's a simple process to clip the child seat in and out.
Front-mount.
Front-mount child seats clamp around the steering column and handlebar area.
As you'd expect, these sit out in front of you.
The advantage with this type of child seat is you have visibility on your child at all times, and it's easier to get on and off the bike.
One downside is weight capacity. These usually max out at 15kg (depending on the brand).
Frame-mount.
Frame-mount child seats connect directly to the frame.
A couple of examples include Kids Ride Shotgun and the Thule RideAlong 2.
The Kids Ride Shotgun option is the best for dual-suspension mountain bikes.
Frame-mount options can be cleanly removed from the frame. There's no rack or large fittings left on.
A small exception is the RideAlong 2 has a small bracket that stays on the seat tube.
The Kids Ride Shotgun option can accommodate kids up to 27kg. The Thule RideAlong 2 is 22kg.