The answer to your question requires a little background on the various types of TiVos:

There are three series of TiVo. The TiVo S1 is a very old series, and is little more than a VCR, with a fantastic program guide and great software for setting up recordings, which are recorded onto a hard drive instead of tape. The TiVo S2 were still analog (NTSC) DVRs, but more sophisticated, with broadband features, such as the ability to schedule recordings from the web, transfer recordings to computers and mobile devices (and to other TiVos), what Internet videos from YouTube, Netflix, Amazon, etc. Both the S1 and S2 can "use" a cable box (changing channels on the cable box via an IR emitter that comes with the TiVo), so you can theoretically record digital cable (as analog video) using these devices. The TiVo S3 and its successor the TiVo HD are digital DVRs. They have six tuners each (two of which can be operated at the same time): two NTSC tuners, two ATSC tuners, and two QAM tuners. The S3 and HD support CableCARD, and that is how you can receive digital cable service using either of those boxes.

A TiVo S1 or TiVo S2 will work like a VCR, so simply put, your cable company won't know about it unless you involve them in connecting it.

You could hook a TiVo S3 or TiVo HD to digital cable, without getting your cable company involved. However, all you'll be able to do is manually tune in channels, and manually start and stop recordings. There is no way to map the program guide data to the channels unless you have a CableCARD. Also, without a CableCARD, you won't be able to decrypt any encrypted channels... that's what the CableCARD is mostly for. (And of course, often expanded basic channels are encrypted.) Essentially, if you're not going to get CableCARD for your TiVo S3 or TiVo HD, my recommendation is to not get a TiVo S3 or TiVo HD.

CableCARD cannot be purchased: They can only be rented from cable companies (and all but a few are required to provide them). Satellite services are exempt from supporting CableCARD and neither of them do so, or in any other way allow you to connect any digital DVR to their system, other than those that they specify are acceptable and support their own proprietary security scheme.

CableCARD, of course, must not only be provided by, but activated by, your cable company.

Please let me know if you want any more details on any of this.