When do I have to perform a move in DDR?

I just attempted DDR for the first time ever today. Sometimes I was sure I was hitting a move at the right time (when the move covered the arrows up the top), but it wasn't counting.

1. How does the timing system work for moves? For example, how much time before and after the notes overlap do you have?
2. Is there a significant delay from you hitting a panel and it registering and does this vary much between machines?

1. You're right, you should be pressing the panel at the exact moment that the arrow covers the arrow at the top. The game gives you a grade based on how well you timed the step. The grades are "Miss", "Boo", "Good", "Great" and "Perfect" (and an ever higher "Marvelous" rating in newer iterations). You want to aim for Great or better. The exact grades and timing windows vary a lot from version to version, and are usually wider on the console versions than they are in the arcade. The timing window for the best ratings (Perfect, Marvelous) is usually on the order of ~20-40ms (source: StepMania wiki), or about 2-4 frames.

   The key to hitting the arrows precisely is stepping to the beat of the song, as opposed to reacting to the arrows. If you're reacting to the note, you're already too late. If you look closely at the arrows at the top, they are blinking to the beat of the song. This will help you feel the rhythm, and on easier difficulties you'll almost always press a panel at the moment the arrow blinks. As the songs get more difficult, you'll have to step on half-beats, quarter-beats, and even more complex rhythms. It helps if you have some music experience, but the clonk of your footfalls should sound even, like a drum playing.

2. Assuming the dance pad is in good condition, there should be no noticeable delay from when you hit the panel to the game's response. If you're playing on a home console, the included plastic mats are usually not particularly good and can break over time, causing unresponsiveness. If you're interested in jumping down the rabbit hole and getting into DDR, it might be worth investing in a better mat. The ones with the foam inserts are a good step up from the cheapo ones without being particularly expensive: DDR Foam Mats

   If you are playing at home, many modern TVs can introduce some delay. Make sure your TV is set to "Game" mode, if it's available. Also, the DDR console versions often have a "sync" setting in the options that you can use to fine tune the timing.

Obviously, you want to time your step as best you can when the moving arrows overlap the stationary ones at the top of the screen. One thing that a lot of beginners have trouble with is that they start moving when they should already be hitting the note. Make sure that you begin moving towards the note early so that you actually contact it in time with the song, rather than being slightly behind the beat.

Generally, because the pads are hard-wired, there is very little or no input lag, and you'll want to stamp the pad exactly when the notes are lined up. One thing that makes this difficult is that many times the pads are not perfectly responsive, and stepping on an arrow will not ensure that you get credit for hitting that note. This is a difficulty I've had any time I've tried playing DDR.

Specifically:

1. You can actually hit the notes out of order. Each note has its own separate timing window irrespective of the notes in other columns. If there are many notes in the same column whose windows overlap each other, credit will be given for the earliest note on that sequence that's window is still open.
2. No, this is fairly constant and instantaneous.