Arn't ferrimagnetic materials usually very hard magnetic (or was it very soft magnetic?). I recall them having uses in specific circumstances at least, though they didn't have much to do with exerting magnetic force. Antiferromagnetic materials are used in Spin valves, which are a primary component of hard disks.
I honestly couldn't tell you if they're usually hard or soft, (hard basically being resistant to changes in magnetisation, so hard to magnetise, but also keeps the alignment for a long time after being magnetised, and soft is the opposite), but what they basically boil down to is materials where the crystal structure prevents all the moments aligning parallel to the field.
In ferri-magnets, the moment forced into opposition is less than those in parallel, so, although the material acts similarly to a ferromagnet, it is weaker than one would expect based on the number of magnetic species in it.
e.g.
In this case, the applied field is downwards.
A basic antiferromagnet is similar, except that the opposing moments are exactly opposed, resulting in no net magnetic field. So,
Most forms of iron rust are actually antiferromagnetic, but above a certain temperature (the Neel Temp.) they transition into ferromagnets (higher still, the Curie temperature, is the point any ferromagnet becomes paramagnetic instead).
Now, I've only gone through the most basic cases here, there are also things like spiral antiferromagnets, where each subsequent moment is tilted up and across a little from the previous, resulting in things like this
where applied field is down the axis of the beads shown. Spiral AFmagnents are cool because, in thin films, they have a net moment perpendicular to the field, but in bulk samples, do not. Of course, now we're starting to get into what my supervisor worked on, rather than what I did, which goes well beyond what lay-discussions are capable of.