Here's what I have so far.Maybe it is time for a terminology lesson.. This is going to be a little long-winded ..
Dvr/tivo = this is a box usually provided by your cable/satellite provider which allows you to "record" programs to an internal hard drive on the unit .. Now there are ways of connecting this to a pc but generally what most people (myself included) do is to connect audio/video cables from this box to either a vcr or dvd recorder (see below) .. When we want to record something that we have "saved" on our dvr we select the program on our dvr and play it, and then hit record on our vcr or dvd recorder. So let's say I recorded the Pens game last night to my dvr, this morning before I left for work I started the game on my dvr and then hit record on my dvd recorder.
Dvd recorder = a device normally connected to your cable/satellite box (doesn't have to be a dvr **) and/or your vcr .. This device may or may not have a hard drive, its primary purpose is to allow you to copy content from the source (cable/vcr) onto either the hard drive of the recorder itself or to a dvd. Why you ask would you want a hard drive, now I have the program on my dvr and on this dvd recorder hard drive?? Because once it is on the hard drive of the dvd recorder you can EDIT it, remove commercials, break it into multiple segments for multiple discs, remove any pre-show or post-show that you don't want. Plus it just makes everything easier :) Most stores do not carry the dvd recorder with the hard drives, they used to but don't anymore. Most people think this is because the tivo people have pressured the retailers not to sell these because for most people they don't record as much as the people I hang out with :) So they would record a program as they watch or schedule a recording (like you do with your vcr). If they would do that they would have no reason to have a tivo/dvr and the sales of those devices would plummit.
There are ways to connect this device to a PC but most people do not do this because the setup for that is a little more intricate and requires software to be purchased on the PC.
** so now you might ask, why do I use the dvr AND the recorder with a hard drive. Well, I only have 1 TV connected right now (really!), so if I were to record directly to my recorder, I could only record/watch 1 program at a time. With the dvr I can record 2 programs at once, and watch either one.
Dvd burner = a device connected to your PC which allows you to burn music, files (for backup), or movies (of course legally!), or copy other dvds. Normally when you buy a burner it comes with the necessary software to allow you to burn the above items, some people buy additional software which allows them to edit audio/video, create pretty menus etc. That isn't necessary to do the basics though.
So .. If you want to copy your vhs tapes to dvd or record programs from cable, you probably want to get a dvd recorder - if you want to buy it in a store, you will not be able to buy one with a hard drive. To record from your vcr to dvd, you would push play on your VCR and record on the recorder; if you wanted to record a Pens game, you would use it the same way as you do now with your VCR, setup a timed recording, put in a dvd and off you go (of course it is a little more complex, than that you have to set your recording quality (same idea as sp/lp on the VCR). Most of the recorders in the stores are fairly cheap nowadays, toshiba & pioneer are good names, some people also like panasonic.
Then if you want to be able to copy your dvds for other people to see or to make backups or whatever, you would need a Dvd burner for the computer.
.... I did simplify some of the options because you are just in the "planning" stage .. Later on, you'll have more questions and we can go back to this for reference and make some changes because things CAN be a little more complicated but it helps to understand the basics... when people say recorder or burner if you aren't following what they say you will be left out in the cold!