Only the DVRs that DirectTV has available for their customers are compatible with the DirectTV signal and equipment. Some DVRs are universally compatible with cable TV and can be used to record programming on those services, the same is not true with any DVR and DirectTV satellite service.
DirectTV's proprietary compatibility is unlikely to change. DirectTV (and satellite companies in general) want their customers to use their proprietary equipment. However, there is good news for customers who want the quality associated with the TiVo DVR platform: DirectTV has re-entered a partnership with TiVo, and they are working together on a new High Definition (HD) DVR that will be available in the near future.
In order to work with the DirectTV signal, the TiVo HD DVR will need to be purchased from DirectTV. This can be frustrating for cable customers who switched to DirectTV, but the bottom line is that the only DVRs that will be available for DirectTV for the foreseeable future will be sold by DirectTV.
DirectTV's DVR has the standard features most DVR's have; that is, the ability to record a show that's going to appear later. It is available in both HD and regular (for those without HD televisions). It can be set as far ahead as the schedule can display, and it can be programmed to record whole series as the shows appear, so there are no missing shows and no need to re-program each week. DirectTV also has some mobile apps that make it easy to program the DVR away from home using a compatible phone or an iPad.