This can be quite a major exercise in its own right. There are lots of different types of testing- hopefully most of it will have been done by the software company, but you should get involved in:

a. Does it do what we want, and does it do what we asked for? You might find that it does what you asked for, but not what you wanted! This is where careful specifications come in. Even if the specification was wrong, involvement in the design and build can prevent many problems at this stage.

b. Does the software misbehave? This basically is quite tough to check, and, for critical software could be much more expensive than writing the software itself. For example, if the software should find a specific person who works for a specific company, but actually only checks for anyone at that company then the software may work perfectly provided that there is only one person at that company. Check it works properly with realistic data, and spend long enough on each area to be comfortable with it.

Ensure that you keep a detailed log of all the problems that you found. You should describe what you did and with what data (eg "searching for Howlett at Data Technologies Ltd found the wrong people") and ensure that you can't reproduce the problem once the company claims to have fixed it.