The trade exception can be used to acquire a player without matching salary.
It is around $10.3 million.
You cannot combine a trade exception with another exception (trade exception, MLE, etc) or a contract. For example, you cannot use it and Courtney Lee to bring back a player making $15m. Since picks don't count as salary, you can combine it with picks (or structure it so that the picks are being traded for nothing).
The trade exception expires one year after the trade, which was finalized on July 12. Since the Celtics are hard-capped due to accepting Bogans in a sign-and-trade, the Celtics are unlikely to use it before the July moratorium. Additionally, the luxury tax is calculated based on payroll on the last day of the regular season, so the Celtics can use it after that point without running contrary to their desire to avoid the repeater luxury tax rate.
The Celtics can take back up to $10.3m in salary. The team can also break it up into pieces and use part of it on one player, reserving the rest for later.
You don't send trade exceptions to another team. The other team may gain a distinct trade exception as part of the deal that has the same one-year deadline from the time of the new trade.