You have two options on how it's run. A server licence allows you to run it yourself (valid for a year). Or you could buy a Cloud system and have Atlassian run it for you (pay monthly or annually), which saves you having to get hardware and backups etc.
Cloud and Server are the same system, but have some significant differences for the users. With server, you can do what you want with it as you have complete access to it. On Cloud, you're limited to the day-to-day admin things. The main reasons people use server over cloud are flexibility, integration (mostly with active directory) and data protection (Cloud is US hosted, this can make it illegal for some uses in a lot of states). Cloud is easier to look after because you can't break it and you don't have to think about the back-end support. Cloud does have "Connect" add-ons, where Server does not, but you can add the more powerful Type-2 add-ons to server.
On the numbers, licences are tiered - you can have up to X active users. The first two tiers are 10 and 25 - sounds like 25 is enough for you.
Finally, Atlassian only gives out free licences to Open Source or secular Charity users. There is a discount for academic use of Server though - https://www.atlassian.com/licensing/purchase-licensing#licensing-2
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