This seems to be a cultural perspective - for me, "for a while" means a longer-than usual length of time. So if you tell me that you'll be gone "for a while", I won't expect you back soon, and I'll go do something else. The only way I could think you would be back soon is if you say "I'll be gone for (only) a short while" - you actually have to say the "while" is short!

a while = any length of time It really depends on context.

for a While== Space of time, or continued duration, esp. when short; a time; as, one while we thought him innocent. and long time ==gap between ur continuity it's like 1,2 ,3,4 or more hour ,for more time gap and just a minute=== it's very closely time between ur continuity very small time

"While" by definition is a period of intermediate length, that is usually short but longer than a minute. How long is a while precisely will depend on its description by adjectives as in ' a long while ago ' or ' a short while ago' . Important is that you understand that 'a short while' can't be 'just a minute', yet 'A long while' is not ages as well; it is still defined inside the boundaries of an intermediate period. When someone is living in a place for years for example, you can't really describe it by ' he stayed there for a long while'. It could be rather used in the context of someone staying for few months in a place. 'While' is also related to temporal or limited time spans marked by a certain action or condition.

It can mean a long time, but for certain "for a while" means that the period of time is longer than "just a second" or "just a minute" especially if the "while" part of the phrase is emphasized. It can mean anything from longer than "just a minute" to a long time depending on context.