There are two types of awards one can receive when a copyright is violated: actual or statutory damages. One also has the ability to recover attorney's fees.
From the U.S. Copyright Office:
The copyright owner is entitled to recover the actual damages suffered
by him or her as a result of the infringement, and any profits of the
infringer that are attributable to the infringement and are not taken
into account in computing the actual damages.
A copyright holder may elect, at any time before final judgment is rendered, to recover, instead of actual damages and profits, an award of statutory damages for all infringements involved in the action.
Statutory damages may be higher than actual damages so there may be a good reason to seek statutory damages instead of actual damages.
The court may also award reasonable attorney's fees to the prevailing party as part of the costs.
However, 17 U.S. Code section 412 states:
In any action under this title, other than an action brought for a violation of the rights of the author under section 106A(a), an action for infringement of the copyright of a work that has been preregistered under section 408(f) before the commencement of the infringement and that has an effective date of registration not later than the earlier of 3 months after the first publication of the work or 1 month after the copyright owner has learned of the infringement, or an action instituted under section 411(c), no award of statutory damages or of attorney’s fees, as provided by sections 504 and 505, shall be made for —
(1) any infringement of copyright in an unpublished work commenced before the effective date of its registration; or
(2) any infringement of copyright commenced after first publication of the work and before the effective date of its registration, unless such registration is made within three months after the first publication of the work.
So, you can only recover actual damages and neither statutory damages nor attorney's fees if the copyright is not registered.
Statutory damages can only be collected for copyright violations that occurred after the effective date of the registration or within three months of first publication if registration is made within those three months.