Federal inmates do have access to a limited email system called CorrLinks, and it works differently from standard email in a few important ways.
The key thing to understand is that your boyfriend has to initiate the connection, not you. He submits a request through the TRULINCS system at the facility, which sends you an invitation to your email address. Once you receive that invitation, you go to CorrLinks.com, create an account, and accept the connection. From that point you can exchange messages back and forth through the platform.
You cannot create an account and search for him or add him yourself. The invitation has to come from his end first, which is why you are having trouble getting started. Ask him to submit the CorrLinks invitation request at his facility and watch your inbox for the email. Check your spam folder as well, since automated emails from correctional platforms sometimes get filtered.
Messages through CorrLinks are not instant and are not private in the traditional sense. They are reviewed and monitored by the Bureau of Prisons as a matter of policy, the same as phone calls and regular mail. There is also a per-message fee that comes out of the inmate's trust fund account, so he will need funds available to send messages.
For state inmates the email situation varies by facility and state system. Some states use JPay for inmate messaging, others use different platforms, and some do not offer email at all. If your boyfriend is in a state facility rather than federal, ask him specifically which platform his facility uses so you can set up the right account.