If you're talking just about some "random" piece you would use "einen". E.g. you need a mirror for shaving, and you don't care which one it is:
Gib mir bitte einen Spiegel zum Rasieren.
If you use always the mirror inherited from your grandpa for shaving, and the listener knows this, you would say:
Gib mir bitte den Spiegel zum Rasieren.
So if your talking about a particular mirror you actually care about, you need to distinguish if this mirror is already known to the listener or not. E.g. if you mentioned it before or if you're holding it in your hand right now (or if it is unique and "common knowledge" thing, like the United States of America) you would use "den". If you introduce something new, it often gets "einen", but after the introduction you would use "den":
Ich habe heute einen tollen Spiegel gekauft. Den (Spiegel) schenke ich
meinem Onkel zum Geburtstag.
If you use "den" in places where "einen" is expected, people often think they missed some part of the conversation - you seem to talk about something they are apparently supposed to know, but they don't. If you use "einen" in places where "den" is expected, people often think you speak about a altogether different thing.