With "TiVo" being synonymous with "DVR," it would not be unreasonable to think the TiVo Mini is a small DVR. It isn't one.
Instead, the Mini is another part of the company's whole-home viewing experience, joining the TiVo Stream, which streams your recorded content on iOS devices.
Similar to the Stream, the Mini feeds off of a TiVo Premiere 4 or XL4 DVR. Connect the Mini to a secondary TV and to your home network, and you'll have access to live and recorded TV, video-on-demand services (for Comcast Xfinity customers, at least), and other broadband-delivered content through the main DVR.
Basically, it gives you a chunk of TiVo DVR functionality without the need to buy an additional DVR and pay for additional services or equipment from your cable provider. But, like all things TiVo, it comes with a rather hefty list of caveats.
One of the nicest things about the Mini is that getting it up and running is simple and doesn't require any calls to a cable provider. The TiVo Mini needs three connections: power, network, and audio/video. Power is handled by a fairly small wall wart, and HDMI is used for AV (component and composite connections are available, too, but require a breakout cable with a 3.5mm plug).
Connecting to your network can be done over a MoCA connection (Multimedia over Coax Alliance) or Ethernet; MoCA, which uses your home's coax cable to stream between devices, is preferred and support is built into both the Premiere 4/XL4 and the Mini.
The Premiere 4/XL4 needs a wired Ethernet connection to your router as well as a connection to your home's coaxial wiring. If your router is nowhere near your TiVo DVR, you'll need to connect a MoCA network adapter to your DSL or cable modem (unless, of course, you have MoCA support built into your modem). You could also use a power line adapter, but, again, MoCA is preferable since it gives you a fatter pipe to meet the high-bandwidth requirements for streaming live TV. Wireless is not an option for live TV or anything else.
The Mini is controlled with a standard TiVo peanut remote that is included with the box. There is also a USB port, which can be used to add TiVo's Slide Bluetooth remote, or, should you decide to hide the box behind your TV, an IR extender. There are mounting holes on the bottom, so it can easily be attached to a wall. If you're looking to extend your TiVo experience into a bedroom or other space where you don't want a large DVR, the Mini is a way to do that.
Once connected, the Mini finds the TiVo DVRs on your network. Then, you point it to the one you want to attach to. Each Premiere 4/XL4 can support live TV streaming on up to two Mini boxes at a time. However, for each Mini, you have to surrender a tuner on the host DVR, which means you lose it for recording. In other words, if you dedicate two tuners to two Mini boxes, you'll be able to record only two programs at a time with the Premiere 4/XL4. You can add more than those two Minis, however; the DVR supports up to three HD streams, so a third could be streaming recorded content, while other units can access apps like Hulu Plus or YouTube.
The option to add more than one or two Minis will become more important when TiVo adds dynamic tuner allocation sometime in 2013. That would eliminate the need to dedicate a tuner to a Mini and, instead, a tuner would only be needed while in use by a Mini for live TV streaming. If not in use, the tuner could be released, freeing it up for the DVR to record to, or for other Minis to use.
This also means that because each DVR supports three HD streams, dynamic tuner allocation will allow up to three Minis to stream live TV. But that's still to come and if you're still confused about what the requirements are right now, the TiVo Mini FAQ page explains the current setup options.
Again, the Mini isn't a DVR, so it doesn't have its own storage. But, it sort of eliminates the need for a secondary DVR since it can control a tuner or two of a Premiere 4 or XL4 for watching live TV and setting things to record. You also have full access to your recorded programs, so you can start watching something in one room and then finish it in another. All without paying any fees to your cable provider for additional services or equipment like a CableCard.