Yesterday I attempted to send two bitcoin transfers from Livecoin and almost 24 hours later it hasn't been successful. When I check the transactions they have 99 and 95 confirmations. What does that mean?
It means that there have been 99 blocks after your transaction that have been succesfully confirmed.
99 blocks have basically confirmed that your transaction is real, and now ( and still does) belongs in the ( bitcoin ) blockchain.
So, yes your transaction has been successful, if you haven't been credited your bitcoin, or received it, something else is wrong. Mind linking me your TXID?
The more confirmations a transaction has, the less likely it will be that it could be overthrown in a 51% attack, or for example a double spend ( 1+ confirmations already makes sure that isn't the case)..
Generally a transaction with over 6 confirmations is seen as "Irreversible", (aslong as there isn't 1 miner with 51% hashpower) since it would be mathematically impossible to throw over these 6 blocks.
See Why is 6 the number of confirms that is considered secure? for more information about why these confirmations are so important.