If the male fitting is 3/8" or less it should have three full wraps of white (single density)teflon tape.

If the fitting is 5/8" or greater it should have 3 full wraps of red (triple density) teflon tape.

When the fitting is tight there should be no more than two treads exposed.

Make sure you have the correct teflon tape, the fitting is properly tightened and not cross threaded.

If it still leaks the most likely cause is a cracked fitting which would have to be replaced.

I am certain the wrapping was in the correct direction. It is a small fitting, and the width of the threaded area is smaller than the 1/2" width of the tape (I had to smush up the edge of the tape so it wouldn't hang over the end of the fitting). Could that make a difference?

Also - I'm not sure I described the leak quite accurately. It is leaking NOT where the threaded parts fit together, which is the junction of the filter unit and the threaded fitting, but rather at the top of the fitting, where the copper pipe exits. Not sure if that makes a difference in the diagnosis/cure.

Does this filter supply an additional faucet or does it supply the cold faucet at the sink?

Since you mention COPPER tubing it most likely uses a compression fitting and you may have kinked or cracked the tubing.

If you don't know what you're doing then the easiest solution is to get someone in there to correct the problem or shut off the water and remove the filter along with the tubing and take it to someone who knows what they are doing and can advise you.

Any help we can provide is based SOLELY on your ability to accurately describe the components and the problem.

You will have to either accurately describe the problem and components or post a picture or move it closer to the screen so we can see it.

If you are intent on replacing the filters yourself the installation might be improved to make that easier for you.

The filter supplies its own separate faucet. I've changed this filter myself several times without incident, and it's pretty simple to do, but perhaps this time something got kinked or broken (the unit is quite heavy when full of water, and there isn't a lot of play in the in/out lines, so it might have over-stretched this time). I don't know the names of all the fittings, but I generally know how they work and have fixed leaks, cleaned and unclogged drains, etc. in our home before - but this one isn't so obvious to me. I will try to post a photo, and might just take it off to show the folks at my local helpful hardware store!

Might be best to convert that install to PE (polyethylene) tubing and John Guest fittings.

In general, I prefer my drinking water to flow through non-plastic, but I am interested to know why do you recommend this change, justalurker.

"In general, I prefer my drinking water to flow through non-plastic"

Why? Do you drink bottled water in plastic bottles?
I guess there's no PEX in your future.

PE is the recommended tubing for ROs and filter assemblies because it is easy to work with, flexible, cheap, and reliable. Using PE along with John Guest fittings makes a filter change a two minute exercise with post job leaks being a VERY rare occurrence.

Copper as you are, I think, going to find can not be flexed and bent without weakening and ultimately cracking or crimping.

Copper works fine when it is installed and left alone.
Copper is not recommended when installing and RO.

The filter change itself is easy, and does NOT involve disconnecting the connection that is leaking now - the filter case splits apart in the middle (at the "equator"), the old filter screws out,and the new one screws in. What takes the longest is taking the whole unit off the hook that holds it to the cabinet wall, since it's a tight fit.

And no, I rarely use plastic water bottles anymore (yes, I realize there is plastic everywhere, but given a choice, I avoid it. Still - if the hardware store staff convinces me to switch, I'll consider it).

Also - what's RO?

When you take the "whole unit off the hook that holds it to the cabinet wall" and unscrew the filter you're putting stress on the copper line and that leads to leaks and failure.

Bear in mind that the water line to your filter and from your filter to the faucet is at house line pressure and when a leak occurs in can be at considerable flow resulting in considerable damage if no one is home to see and stop the leak.

RO = Reverse Osmosis.

RO water is typically so clean it is aggressive towards piping.

Plastic has the best resistance to being attacked.

The very large and cross linked molecules do not come loose as easily as copper atoms, keeping the water cleaner.

Where very high purity is needed, aluminum cladding on the outside of PE is added.
This limits the ability of air to enter through the tubing.

This is not a reverse osmosis filter.
I called multipure and they sent me a brand new piece - some sort of proprietary (they claim) compression fitting. I'm ready to do the replacement, but now can't get the old one off! I've already disconnected the section of pipe from the flex hose under the sink so it's easier to work with, but no amount of tugging will get it loose (when I took it to the hardware store looking for a replacemtn, it came apart easily, but I guess I put it back together too well - still leaks, though). Any tips?