Leviton is a manufacturer. They make all kinds of electrical equipment, including both simple and smart switches. If you are installing a smart switch of any type (timer, WiFi, remote, etc.) then we'll need to know the model # and possibly additional specifics to get this figured out. But assuming it is a simple 3-way switch replacing the existing switch, this is straightforward.
As noted in comments, you should always remove wires that are screwed into a device using the screws rather than cutting the wires. That being said, you have one advantage over some other questioners: you know 100% for certain which wire was connected to each screw!
The trick with replacing 3-way switches is figuring out which screw terminal is for which function. There are (excluding ground) 3 connections:
- Common = Hot or switched hot - The actual function depends on whether this is the first or last switch in the series of panel -> 3-way switch 1 -> (possibly 4-way switches) -> 3-way switch 2 -> light (or other switched device).
- Travelers - There will be two of these and they are functionally identical.
The screws can be configured "any which way". Typical would be "common on one side, travelers on the other side" or "common on top, travelers on both sides on the bottom", but technically anything is possible. Fortunately, it is easy to figure out:
- Beg/borrow/buy a multimeter if you don't already have one.
- Set it to resistance or, as most have, continuity testing mode. Depending on the model, continuity testing may be combined with the any/all/lowest resistance (Ohms) mode.
- Check continuity with each pair of screws. Flip the switch. Check continuity with each pair of screws. If your multimeter beeps for continuity (technically, resistance below a very low threshold) then you can do this without even looking at the meter.
You will find that one screw is part of a continuous pair no matter which way the switch is flipped - that is common. The other screws are each part of a continuous pair only in one switch mode - those are travellers.
On the new switch, you can do the same testing, but it is easier to read the instructions. In addition, most new switches will have different color screws (e.g., black, brass or silver) - the two screws that are the same color will be the travelers and the other screw the common.
- Connect the old common wire to the new common screw.
- Connect the old travelers to the new traveler screws.
That just leaves ground. Most new switches will have a ground screw. If you are using a metal box and the switch has a metal yoke then physically installing the switch (provided there is no paper, paint, rust, dirt, etc. blocking solid contact between yoke and box) takes care of ground. If not, if you have a ground wire available in the box then you should connect it to the ground screw. And if you don't have a metal box and you don't have a ground wire available then, as I understand it, for simple switches (smart switches might need ground) you can ignore that ground screw.