Ideally you have a stored procedure adding the assignment to the table - and if so you just need to modify the stored procedure to send an email when an assignment is added.
If you don't have a stored procedure performing the insert (why not?) then your options are:
- Service Broker
- Agent Job
I would not recommend using a trigger - to do that you need to record the information in a separate table that works like a queue and have a job process the queue. That is essentially the same as using an agent job - where the agent job is scheduled to run every xx minutes - pulling any new records - then using a cursor over the result set you call sp_send_dbmail.
With a service broker - SQL monitors the table and adds rows to the queue - then processes the queue. However, this requires a lot of time and effort to setup and build.