I have a kitchen light and a living room light (ceiling lights) that have their switches side by side (two single pole switches). The kitchen light began not coming on sometimes when I flipped the switch (but would come on after several off/on tries). I replaced the switch with a new single pole switch, and I thought I wired it the same as before, two black wires on the top brass screw (ON), and a third black wire on the bottom brass screw (OFF). I did not touch any other wires, only the wires from the old switch. I tried the kitchen light (with the new switch) and it came on each time I flipped the switch to on. Next I tried the living room light (with its same old switch that I did not touch), and it came on every time I flipped its switch on. I thought everything was now fine, but then while I had the kitchen light on, I tried the living room light again and it did not light up. When I turned off the kitchen light, I could then turn on the living room light, but ONLY if the kitchen light is switched off. Somehow the Kitchen light being switched on, now blocks the living room light from getting power?

Sounds like you got confused as to which is the hot leg. Go get some white vinyl tape and a sharpie and test the leads to figure out what they are. Also, determine if your living room and kitchen lights are on the same breaker or separate. If on the same breaker, you need to be sure you have the hot lead going to each switch having their own switch legs going to the lights.

In this situation it's best to strip it all down and start making determinations as to which lead is really what. You could have a 3 way switch involved with either the kitchen or the living room and what you think is hot isn't really hot, but traveler leg.

Yes, sounds like you hooked up to a traveler wire on a 3 way. Not an uncommon thing to do.

What you are telling us is not logical. Not that I don't believe you but it is just not logical.
If you said that the living room light only worked when the kitchen light was ON, that I could accept. But to say that it only works when the kitchen switch is OFF is not logical.
Describe all the wires in the box. Are these wires in cables or are they in conduit.

You say that you only disconnected the three wires to the kitchen switch. That you did not touch or change any other wires. Is that correct?

You have three wires. Lets call them A, B and C. Let also say that you currently have A and B connected together under the top screw. Which screw does not really matter, the current goes through the switch in either direction. What matters is which wires are connected together.

So you have A and B connected under one screw and C under the other screw.
Try A and C under one screw and B under the other.
Then try B and C under one screw and A under the other.

Those are the only three possibilities.

I suspect that there is something else wrong (done by someone else).

Describe the wires in the box and if possible pull both switches out of the box and post a picture.