This might come up when you apply for a green card, as you will be fingerprinted and those will be searched through all databases, including interpol, etc. for possible matches.  If it does come up as a hit if you have ever been arrested or charged, you will have to provide an explanation as to the nature of the offense and the disposition thereof.  If it is something that does not make you inadmissible to the US, you will be granted permanent resident status; if it is something that makes you inadmissible, you may need to deal with that further, by way of waivers, etc.