What is the reapplication process for a suspended dispatcher certificate?
I know very little about this, but JO 2150.3C says that if a certificate is suspended for a fixed period of time, it's reinstated at the end of that time if you meet whatever conditions the FAA requires (p. 7-4):


  Indefinite suspensions are used when the Administrator has reason to
  question, but is unable to determine, a certificate holder’s
  qualifications, or when the certificate holder does not comply with
  statutory or regulatory requirements to cooperate with the FAA. For
  indefinite suspensions, the Administrator suspends the certificate
  until certain conditions are met, e.g., until a reexamination or
  reinspection under 49 U.S.C. § 44709(a) is successfully completed. The
  period of time the certificate is not effective is specified in the
  order of suspension, and once the required period has passed, the
  certificate is reinstated.


Your question doesn't have many details - and I understand that you might not want to share them! - but I take that to mean that if your certificate was suspended for one year, it will be reinstated after one year has passed, as long as you've done whatever the FAA wants you to do in order to qualify for reinstatement.

The phrase "until certain conditions are met" implies that there's no standard way to qualify to reinstatement; it probably depends on the reason that the certificate was suspended in the first place. Page 9-15 of the order lists reasons for suspension and they're quite varied, so presumably the "certain conditions" are too.

On the other hand, if a certificate is revoked then it cannot be reinstated (p. 7-4):


  Unlike a suspension, a certificate that has been revoked cannot be
  reinstated. A person whose certificate has been revoked may be issued
  a new certificate provided that the person meets the qualification
  requirements for the new certificate. To be issued an airman
  certificate following revocation, an individual must retake all tests,
  whether written, oral, or practical.


Your local FSDO could probably confirm that, although if your situation is more complicated then an aviation lawyer might be more helpful.