Intervals are all about riding at a defined level of effort for either a given distance or time. Usually, that's bleed-from-the-eyes hard for 1-8 minutes.
Interval effort is defined differently with different tools (HRM vs power meter), and for different durations. Intervals are usually defined by an effort for a time, but can be bound by a distance (like in hill repeats). I have an HRM, but during intervals, I use it for data rather than pacing. I guage my effort based on feel (from experience). All of my intervals are about putting as much real estate under my tires as I can for the defined duration. When you get into longer intervals, the HRM becomes more useful, but in the 1-minute range, your HR is climbing the entire time, so you can't use it for pace.
For a 1-minute interval (my favorite), always do them on a familiar course (I have two courses where I do all my 1m intervals). I start them on the same landmarks every week. My start land marks, for example are the first 45mph sign on Burleson Rd, then the fire dept sign on Mc Kinney Falls, etc. There are 2nd landmarks that I'm trying to get to before my time runs out. These are always slightly too far for me to reach.
I go at them so hard that I can barely ride when the interval is over. It's completely consuming. Then I rest for 3 minutes or so, and go at the next start landmark. My usual route has 7 intervals, with an optional 8th. I've done as many as 10, but that's really rare. I stop doing them when the quality goes down and I really start to slow -- at that point, the damage is done, and it's time to recover.