Recently I got terminated during my probation period. Does this mean that my next employer can do a background check on me to find out if I've been terminated if I sign the form that allows the company to do a background check on me? Will I have a lower employment chance if they find out that I got terminated then?

Well, if your next employer calls your previous employer, they'll find out for sure.

After that, awkward questions will follow.

Whether you get the job or not depends on how best you answer those awkward questions.

How does the prospective employer find out about the performance and character of the candidate and then make decision to hire if he is not allowed to call up his current employer?