Which software is best for editing the audio of a video?
Depending on what sorts of editing you want to do, you might be better off doing your audio editing within your video editor itself. I have used Premiere and Final Cut Pro, and both have fairly serviceable audio editing and effect processing support. For more detailed audio editing, Logic Studio has excellent support for editing audio with a reference video, with the caveat that it works best with complete video segments, and only a single video segment at a time (ideally you'd be using it for foley or soundtrack work or the like on an already-fully-edited video, or at least one where the timing isn't going to change).
What to look for is either a video editor that allows you to apply some sort of effect plugins to your audio tracks, or an audio editor that allows you to pull in a video track, extract the audio from it, and display the video on your timeline and shows the playhead's video frame.
I have not done any audio editing for video, but I have done a bit of recording music.
I have used Audacity in the past and find it good for simple multi-track recording. Audacity is easy to use for basic recording and editing and it is free.
If you need to do anything with effects or more complex editing I would recommend a lower cost DAW such as Reaper. Reaper will allow you to record and edit audio. You can add and adjust effects in real-time. Reaper has excellent documentation to allow you to get started recording and editing quickly.
If you really need a super high powered ultra professional DAW you can get Pro Tools. I have not used it personally, but I have heard from a few people that there are some really great tools for making audio editing much easier. The only downside of this program is that it is fairly expensive.
Once you have edited the audio, you should be able to re-integrate it into your video using whatever video editing software you are using.
I've had positive experiences with Steinberg Nuendo whilst doing audio for visual work. It is very similar in operation to Cubase and allows use of everything you would have access to in a standalone audio production (VSTs, automation, etc...)
Further to this there is an in built video playback function such that you can check the levels, timing, etc of your audio alongside the raw video footage in real time.
It all depends on what you want to do with the editing software.
Your best bet would be to stick with the audio editor built into the video software since it will keep the audio lined up with the track, although it wouldn't be very difficult to do even after using any of the programs I'm about to mention.
If its bare minimum things you could get away with using Audacity, but I wouldn't recommend it.
If you have the Adobe Creative Suite, it will come with Adobe Audition which is a very good program. I have personally used it to remove vocals from audio tracks, which it worked great for. It's interface is a little overwhelming at first, but like any DAW (digital audio workstation) it just takes some getting used to.
The industry standard for audio recording, production, and editing is Pro Tools. Almost any song you hear played on the radio or on a commercial nowadays has been recorded and produced in Pro Tools. If I had to recommend a program it would be this one, Pro Tools.
You could really get away with any DAW though. Personally, I like Ableton Live the best but for what you are describing, it may not be best suited for your needs. You could even get away with using FL Studio as long as you know what you're doing... A lot of people rag on FL Studio but I used it for a long time and it can do anything any other DAW can do although I've never liked the way it records audio or its midi interface. Its ease of use is what draws a lot of noobs to it, but I can't say it's bad.
But to summarize and pick one choice, again I would go with Pro Tools.
If you only need the basic audio editing functions (such as extract audio from video, remove unwanted part of the whole audio, convert ogg audio to mp3, etc.), you can use professional video converter. The number of separate audio editing software is not much, but most of the video converting and editing software can also edit audio perfectly. For Mac OS, you can free trail this audio/video software for Mac. For Windows, this audio/video software can help you edit audio with ease. PS I use the first one for Mac, and the second one was recommended by others on facebook. Hope it helps.