The passphrase memory you actually want
If your private key has a passphrase (and it should), you'd otherwise type it every single SSH command. ssh-agent is a small daemon that holds your decrypted private keys in memory: type the passphrase once, the agent uses the key for the rest of your session. On macOS, you can layer the system Keychain on top of that so the passphrase persists across reboots, too.
Using ssh-agent
On macOS, ssh-agent is auto-started for your login session. On Linux, your desktop environment usually starts one — if not, eval "$(ssh-agent -s)" in your shell does it.
Agent forwarding (-A) and its trap
Agent forwarding (-A) lets a remote machine use your local agent — handy when you SSH into a bastion and from there need to reach an internal server with the same key. The trap: anyone with root on the intermediate host can use your forwarded agent socket to authenticate as you to anything else.
For hop-through scenarios, ProxyJump (Track 3) is almost always safer. -A only between machines you fully trust.