How to make the volume of a video even
Uneven sound is evened out in Cut by Zaweba by normalising to the EBU R128 loudness standard, so a video plays at a steady, comfortable level.

For a video whose sound is too quiet, or that jumps between quiet and loud, use normalise in Cut by Zaweba. It applies FFmpeg's loudnorm filter to the EBU R128 standard, the loudness measure used by broadcasters, so the clip plays at a steady level without you guessing at numbers. It's in the Filters tab beside the volume control.
Cut by Zaweba is free software under the EULA, works offline and needs no account.
Volume versus normalise
These two do different jobs. Volume turns the whole clip up or down by the same amount. That's right when a recording is simply too quiet or too loud all the way through.
Normalise measures how loud the clip sounds to a listener and brings it to a standard level. EBU R128 measures loudness the way people hear it rather than by the single loudest peak, which is why it suits speech, music and mixed recordings alike. It's the better choice when you don't know how far off a recording is, or when several clips should sound alike.
Turning on normalise
Open the video in Cut and play it with Space. The sound comes through your speakers, so you can hear what you're dealing with: quiet speech, a loud burst of laughter, music that drowns out a voice.
Trim anything you don't need. Mark in with I and out with O, or mark a stretch and press Delete. A loud bang in a part you're cutting anyway shouldn't decide the level of the rest.
Go to the Filters tab and turn on normalise. If the result is still a little lower or higher than you'd like, the volume control is right there.
Then open the Convert tab and pick a preset by purpose, such as For WhatsApp, For YouTube 1080p, For email, Small file or Best quality. The estimated size and encoder appear before you start. Only the kept parts are converted, with the filters applied, so the sound is evened out in the same pass as everything else.
What to expect from levelling
Normalise every clip that will be watched together. A set of holiday videos recorded on different phones will vary a lot. Normalising each one brings them to a similar level, so nobody has to reach for the remote between clips.
Don't expect it to fix everything. Normalising evens out loudness; it doesn't remove wind noise, a hum or a voice recorded from across a room. A distant voice will be louder after normalising, but so will everything else that was in the room.
Check the start and end. A clip that begins with a door slam or ends with a cheer can sound abrupt once levelled. A short fade on the sound, also in the Filters tab, softens both.
Saving just the sound
If you want the audio rather than the video, perhaps a talk or a song from a recording, Cut can save the sound of the kept parts as MP3, WAV or FLAC. The sound filters are applied there too, so a normalised clip gives a normalised audio file.
A folder of clips at once
Set normalise once and add many files from the Batch tab. Each gets the same filters, the jobs run in the background with progress bars, and the queue is saved on every change, so it picks up again after a restart. On older PCs, Cut's low-spec mode switches on by itself and runs one job at a time.
Uneven sound is one of the most common complaints about home videos. In Cut by Zaweba it's one setting.