If you've read Shot97's excellent post on Getting the Amiga 500 Online - The You're Not Stupid Guide or watched his video, then you've seen and heard ample mentions of A-Talk III.
If you aren't familiar, in April, 1988, AmigaWorld magazine wrote that "A-Talk from Oxxi... is one of the most powerful telecommunications packages on the market... A good package with few flaws, A-Talk [iii] is probably the most complete telecommunications program available." In the same article they went on to say, "Terminal emulators are A-Talk [iii]'s forte."
Maybe you've got an Amiga online already and don't want to browse (painfully) on the internet, but want to hit some BBSes instead where a lot of social interaction still exists. Yes, BBSes are quite in vogue these days, believe it or not.
Or, maybe you just want an Amiga terminal program and aren't sure where to start. There are few that stack up to A-Talk III in terms of intuitiveness, user experience and depth of capability. The software had contributions from Amiga heavy-weights like Dave Haynie and Robert J. Mical, if you need further convincing.
A-Talk III was developed by Felsina Software in the late 1980's, and was published by Oxxi, Inc.
As of 2010, the software was permitted to be shared publicly by Marco Papa of Felsina Software.
I happen to have found a full-copy of the original Disk 1, with complete HD-install routines.
A-Talk III (Complete full-disk ADF)
A-Talk III Manual
It's worth noting that A-Talk III was later ported to Windows NT in 1995 (which required 16MB of RAM, while the Amiga (I think) needed 200KB).