iMessage, which is Apple's own internet messaging service, can send fairly long messages (several or more thousand characters at least, but I don't know what the upper limit is - may be file size and not character limited). I'd be surprised if there was not some hard upper limit though, as at some point, Apple would want people to use email, and not an instant messaging service intended for short rapid communications.
SMS texts originally were strictly limited to a 160 character limit. Beyond that, the only way to send longer texts over SMS is to break the message up into multiple messages, send each one, and re-combine them on the receivers handset. That's because the bandwidth used by SMS is a small, unused fraction of the cellular voice channel and it is only so big.