"How many students are there in your university?"
Translation:Quanti studenti ci sono nella tua università?
is this sentence without "ci" still grammatically correct? If so, what is the difference
No. "Ci" means "there". How many students are there in your university. English can do without the 'there' here because of the 'in your university', but if you take those words away, you need the 'there'. Italian doesn't think you can leave the 'ci' out because of those extra words (and when you think about it, rightfully so).
To those who speak Dutch: 'ci' = 'er' (when it is not 'us'/'ons'). Hoeveel studenten zijn er in jouw universiteit. Realizing ci=er really makes a lot of translations much more simple. Pensarci is for example _er_over nadenken. English doesn't have a similar word so when learning from English 'ci' is harder to understand.
alievi also means students so why did it mark my phrase as wrong based on that word?
Well, you did misspell it, but I doubt that's the problem. :-)
Apparently anyone who studies something is an allievo, even if he/she isn't going to school. Use this to ask how many students are on Duolingo, though!
Alunno should also fail, since it's not used for college students.
There's more, if you're interested: http://forum.wordreference.com/showthread.php?t=608617