I assume you checked your own criminal history for a reason like education or employment. If so, you may want to be in touch with counsel from your area to see if those two cases are eligible for expungement or sealing in your jurisdiction.
In Illinois Nolle Prosequi means that the criminal case has been dismissed. The term is Latin which roughly translates to "No More Prosecution." In terms of a criminal record, if this is all that is there, you could factually state you do not have any convictions for any felonies or misdemeanors.
You may wish to explore having the entire case expunged. This would have the effect of having this case entirely removed from your record. At my website www.ZippToCourt.com, I have a Zipp To Legal Research page which you can find here: http://zipptocourt.com/Zipp_To_Legal_Research.html
On that page, you will find a link to a guide for expungement in Illinois.
This answer is marginal legal advice and does not constitute an attorney-client relationship. Every client and case is unique. The best advice is to always consult with an attorney. Free legal resources at www.ZippToCourt.com
No. It means the prosecutor has decided not to proceed with the prosecution of that particular charge. Usually, a nolle pros occurs when some charges are dropped pursuant to a plea agreement. If you have had charges nolle prossed, you should ask your attorney to file a motion to expunge these charges from your record. In many states, nolle prossed charges do not magically disappear from a criminal record on their own.
As my collegaues have all pointed out, a "nolle prosequi" is a dismissal on motion of the prosecutor, not a conviction. If these are your only arrests you should look into having the arrest and court records expunged. It sounds as though you may be eligible.