"HBoot" mode is something of a misnomer, really, that I usually see when people mean "reboot into your HBoot menu". I think the reason these got conflated to each other is because
Fastboot mode activates by default when you land on HBoot on most (possibly all) HTC phones. They do exist separately, though, as devices without an HBoot menu can still have a fastboot mode. This is definitely a little confusing, though, since what's really happening is that you're getting rebooted into HBoot which is then immediately loading Fastboot. You should be able to get to the bootloader's main screen by selecting
Bootloader from the Fastboot screen.
Essentially, though, HBoot is just a bootloader, which gives you the ability to chain-boot into several other partitions or modes (including fasboot). I say it's not really a "mode" because it doesn't really allow for any interaction, it just loads and waits for you to choose something else from the menu (with one exception, see below).
HBoot also has the ability to install a properly named and signed zip file if placed on the root of your SD card, which it will attempt to do right after the bootloader has fully loaded. This seems to have been designed as a sort of last-ditch reflash method, and used to be a part of several manual rooting methods on HTC phones (which is now largely unnecessary). The exact naming convention depends on your device, but they're usually of the form
Fastboot mode is the state your device goes into so that it can accept fastboot commands, which largely deal with manually flashing partitions on the device. Notably, you can flash unsigned partition images when in fastboot. It's handy for recovery purposes, but also useful for development and generally hacking around (depending on what you're doing). You can read a bit about the details of fastboot here.
Your "recovery mode" (or recovery partition) is different from HBoot. It is a separate partition on your device which contains a very basic kernel and GUI that can be used to repair and/or restore your system, with the features varying depending on the exact recovery you are using. Typically a manufacturer will ship their devices with their own "stock" recovery, but there are many custom ones available which are more robust. There is a related Android Enthusiasts question which also discusses this.