Can a person be convicted of the same crime in both civilian and military court?
Yes, the military may try someone under their own rules, even after a state court trial. The reason is that military and civilian courts are fundamentally separate systems with their own sets of requirements. As a result, the constitutional right to no double jeopardy or no double punishment is preserved by having separate trials in the two different systems. Both the federal government (which usually represents military interests) and a state may prosecute someone for the same conduct.
Separate Court Systems
The separation of court systems is more reflective of the differences in state vs. federal law than it is differences in the actual trial courts. For instance, local and federal officials often work together, typically selecting the court system that is most heavily involved in/affected by the prosecution. This type of cooperation and court selection can occur when, for example, civilians are arrested on a military base, or visitors to a military base commit an assault against civilian employees.
While US Attorneys usually prosecute felonies committed on base, there is a special system allowing for misdemeanor prosecution of these civilians through the US Magistrate’s Court. In these misdemeanor cases, it has been very uncommon to prosecute the same people for the same actions in both state and federal courts.
Double Jeopardy in the Military Court System
Military members also have special rights when they are arrested, including rights against double jeopardy. Though that may sound somewhat contradictory given the possibility of a double trial, the right against double jeopardy is not a right against participating in more than one court system. In fact, even civilians who were acquitted in state court could be tried for the same actions in federal court.
There are also important reasons why two separate courts sometimes need to hold a trial for the same action. In one high profile case in Washington, a soldier refused to deploy to Iraq and was tried - twice - before military courts. His lawyers argued that being tried twice was grounds for dismissal (double jeopardy), but the appellate military judges disagreed. It wasn't until the soldier's case was eventually argued in a civilian federal court that his case was successfully dismissed on double jeopardy grounds (in this instance, the civilian court had more authority than the military court to address the scope of double jeopardy - the deciding issue). This type of flow pattern through the trial/appeal/military/civilian courts is not always available or even necessary; The details of your situation will always turn on the specific facts, circumstances, and matters of law involved in your case.
The military does not usually punish someone for a crime for which they already received/completed punishment by a state court. This is for practical, legal, and public policy reasons. A civilian conviction, for instance, can simply be placed in the member’s personnel record, result in suspended privileges, or require alcohol or drug screening. Military punishment for an equivalent act is often superfluous. The military often considers such convictions to be administrative matters, and these are rarely assessed as disciplinary or criminal penalties.