What does it mean when a mission is illegal?

It involves performing actions that are illegal and will get you bounties if caught, like attacking Clean ships in non-Anarchy systems or transporting cargo that's illegal in certain systems (such as Performance Enhancers and Slaves).

The mission itself won't get you a bounty, but completing it just might cause you end up with a fine or a bounty on your head if you're not careful.

I don't have enough rep to comment, so adding an answer to correct Cmdr Freiheit's answer.

It is possible to be scanned at an outpost, I've had it happen twice, one time whilst carrying illegal cargo.

In both cases the outposts were aligned to the Alliance, but I'm not sure if that means that there are no Federation or Empire outposts that will scan you. In both cases there were a lot of contacts on my scanner unlike usual approaches to outposts where there are generally none (I play in private groups so I rarely encounter other human commanders).

Getting past security at a patrolled outpost is a lot more difficult than at a normal station. With a normal station if you start to get scanned you can get through the letterbox/entrance quickly enough to break the scan. This is not possible at an outpost and in my experience it is often a lot harder to find or approach your landing pad at an outpost.