As per canon law, Catholics are required to be married before a member of the Catholic clergy authorized to perform marriages. (This does depend on the Catholic rite. In the Latin rite, priests and deacons are permitted to perform the marriage ceremony. This is because in the Latin rite, the husband and wife confer the sacrament on each other. In eastern rites, only the priest is permitted to perform marriages, because the priest confers the sacrament on the couple. Generally, a Latin rite Catholic who marries an eastern rite Catholic must be married before a priest for the marriage to be valid, due to this requirement.)
Now, if one party is not Catholic, there are one or two dispensations that are required. If one party is Catholic and the other is not, a dispensation for disparity of cult is required. If the parties want a valid marriage but do not want to marry in a Catholic ceremony, they need a dispensation from canonical form. Both of these must be approved by the chancery office for the diocese, at least where I live.
If these dispensations are not sought or granted, the marriage is invalid. Now, in order to have the marriage declared valid, the OP needs to speak to his/her priest. Assuming there are no impediments, there are usually two options. One is to have a simple convalidation. This is the Catholic marriage ceremony. It should not be a repeat wedding; the couple can simply make an appointment with their priest, ensure they have valid witneses, complete the required marriage preparation (depending on the situation and recommendations of the priest), complete the prenuptial investigations required (again, determined by the priest), and have a simple wedding ceremony. The second option is a radical sanation. This translates as “healing at the roots”, and may be preferred in cases where one spouse does not wish to have a second ceremony and/or believes their original consent was valid. In this case, the marriage is retroactively made valid as of the date of the wedding, through paperwork. (In the case of a simple convalidation, the couple’s marriage date is the date of the convalidation. It’s a Catholic wedding, although it should not be celebrated with the usual grandeur of a wedding.)