Firstly, it depends what kind of offences you have been convicted of. It would be impossible for a person with a recent conviction of assault to be granted licensure, except in some extenuating circumstances. Fraud, robbery and theft are also seen as being serious in the eyes of licensure organisations. Traffic offenses are irrelevant to licensure. DUIs can and drug-related charges generally have to be relatively recent to be of relevance, unless you happen to have a long history of drug/alcohol-related charges.
Secondly, the period of the offense is relevant. A criminal record check should only cover about 10 years or so (varies from state to state). If the offense is just within that limit, often they will allow registration, although sometimes they will only grant conditional registration, which will restrict your duties.
If the offense occurred more recently than that, you will need to show your nurses board that things have changed in your life during this time, and evidence is required. Such evidence is, for example, a letter from a doctor or psychologist, addictions counsellor, AA facilitator or rehab manager.
The third factor that is relevant is the circumstances under which the offense was committed. If you were drunk at the time and you can show you have been dry for 3 years, you have a pretty good chance. If you had a psychotic episode at the time but are now successfully managed and well-monitored by a psychiatrist, you also have a good chance of beeing granted licensure. There are studies which show that an antidepressant has caused violence and a flattening of feeling towards others, and cases where this was used to mitigate a sentence in people convicted of murder and assault. It won't be long before this defense will also be used in licensure applications.
For this reason, if you are on a good behaviour bond or if you were found guilty but conditionally released "without conviction", the nurses board will want to see the conditions of your bond, to give them an idea of your suitability to practice as a nurse and what conditions (if any) they may need to impose on your license.
If the question on your application is "do you have any convictions" and you were either released without conviction you may answer "no". If the question was "have you ever been charged with a criminal offense" or "had any offenses proven against you" the truthful answer is "yes".
Many states' privacy laws allow a person with an outdated conviction to refrain from disclosing that conviction. For instance, if the conviction is over 15 years old and there have not been any further convictions or current charges, it is considered "spent" and does not need to be disclosed. In the case of a release without conviction, sometimes the finding is considered "spent" when the period of the bond has been served, so even if the question was "do you have any offenses proved against you" you can answer "no". If either of these apply to you, check the law in your state and in other states.
Finally, if the board decides you may not be able to be granted a license, they must give you a hearing. In that hearing you can argue why you would be suitable to practice as a nurse, what changes you have made in your life, and other relevant factors such as those I outlined in the beginning of this post. You can also have a lawyer represent you in such a hearing.
Hope that clears things up for you.