there are a lot of threads here on annulment, do a search, because some people have been giving very good answers. A priest does not “give” an annulment, the tribunal of the diocese investigates the circumstances which pertained at the time of the marriage (not afterward) and determines if in fact the conditions for a valid marriage were present. If not, they issue a decree of nullity (there is really no such thing as an annulment) which says the marriage is null, that is, it never existed. The judgement can be appealed to a higher court.
This decree does not affect the status of any children the couple has. If there was not marriage in the first place, both parties are free to marry if there are no other barriers, and if the circumstance that rendered the first marriage invalid is taken care of. If the first marriage was invalid because of incapacity to make full consent, if that incapacity still exists, the person may not marry, for instance. The tribunal judgement will address that type of thing.
EAch marriage situation is unique, one of the parties should approach the parish priest and ask how to begin the process. No answer here about a specifica case would be helpful.