Time-shifting is what most of us have done for years with a VCR - you record a show you really want to watch, and then watch it later. Media Center can do this, as well as record multiple channels while you watch something else. With the built-in program guide, you can tell Media Center the shows you like, and it will record all instances of it automatically.
This "watch it later" effect is called time shifting. You're still watching the show. You're just shifting the time you saw it. In a major improvement over VCRs, systems like Media Center (known as Personal Video Recorders, or PVRs) allow you to fast-forward and rewind with greater control. You can also pause and rewind live TV. Recording multiple channels means you can watch House and Spicks and Specks, and watch House later when all channel 10 has to offer is Big Brother (which surely no-one reading this watches).
To give a real example of these advantages, one need only look at TV ads. Most advertisements are in standard 30 second lengths. This makes it easier for networks to syndicate content around the country while playing locally specific commercials; with a button on Media Center configured to fast-forward 30 seconds with a single press, skipping ads has never been easier.
Aside from the distress of viewers more easily avoiding commercials, advertisers are also concerned about the effect of time-shifting. Media buying and placement is as much about strategy as deal-making, with placement carefully timed for a sale starting tomorrow, or to suggest pizza as an answer to the hunger pains on a Sunday evening when you can't be bothered to cook. Even if you watch the ads (which is unlikely), they might be advertising an event that has already passed.