Well I can tell you 0 confirms means just that, that there have been zero confirmations of the transaction and it hasn't been included in any mined blocks.
I would wait until you have at least 1 confirmation before sending anything, even with low amounts.
If the transaction amount is significant, I would wait for the number of confirms in proportion to the amount of funds involved.
Personally, I would do something like:
> 1 BTC - 1 confirm is probably plenty
1 - 2 BTC - 2 confirmations
2 - 5 BTC - 3 confirmations
5 - 10 BTC - 4 confirmations
10- 20 BTC - 5 confirmations
20+ BTC - 6 confirmations
Adjust this scale to your level of tolerance for loss and the transaction's sensitivity to time, but again at least wait for 1 confirmation. It is generally accepted that 6 confirmations is pretty secure, but this level can take well over an hour at times.
I should also add that if the person sending you the Bitcoin added a low or no fee when sending, the first confirmation could take quite some time. I have had a few low fee transactions take up to 6 hours to get their first confirmation in recent months due to the amount of transaction backlog and miners starting to refuse to include low or no fee transactions in their blocks.
If the person who sent you the BTC is complaining or rushing you and you haven't seen 1 confirm yet, verify with them if they included a proper fee. You can also look up the transaction details as odolvlobo suggested using a block explorer, such as
Blockchain.info, and it will show you the fee that was included, if any.