Specifically from recordings that other people made.
If you don't have access to a DAW with noise-reduction plugins there is also standalone software designed specifically for this purpose. Audio Cleaning Lab by Magix is a popular one.
Another option is to use a low-pass filter to roll off the very high frequencies where this noise is usually found. However, it's usually difficult to completely remove the white noise without giving the audio a dull muffled sound.
The bad news is that you can't remove white noise from a recording completely. White noise by definition lives all across the frequency spectrum, and can't be distinguished from signal where they both exist.
The good news is that you don't have to. For the most part your brain can't hear noise when it is masked by a real signal in the same frequency range. So the noise reduction approach has two parts:
where there is no audio signal ("white space" or silences in the recording) just turn the gain down and you remove noise completely. This is "noise gating" and can be done by common plugins.
where there is signal, use spectral analysis to find out where in the frequency spectrum your desirable signal is, the use equalization to turn the gain down in parts of spectrum where there is nothing but noise. This is one of the main things "denoising" plugins do.
if you can identify non-white noise elements (hum, pops/clicks) use surgical dynamics and EQ to remove them.
That's the basic forensic audio workflow. It's a hard problem; good luck!
Audacity has a decent noise removal plugin. It's free!
There are plugins that can do this. As far as I know, the RX Denoiser from iZotope is doing a fabulous job. See http://www.izotope.com/products/audio/rx/
You might be able to use an MP3 encoder for this. Experiment with encoding to MP3s at lower bit rates than you normally would, and it should remove much of the noise while keeping the signal, in a psychoacoustically good way. It will degrade the quality of the recording, but it might be acceptable in a mix with other things.
If you are dealing with the white noise created during video production there are several steps to consider. Firstly, the power adapter will bring noise to the project. I recommend using the Camcorder batteries, which add almost no extra noise. The high end Panasonic video cameras have built-in noise reduction filters, which also reduce the noise.
Is this a presentation? Why record the sound in stereo when there is only one voice? This will add echo, which will cause production problems. It is normal to record in mono - it's how the music industry record your music collection. I use a mic that I bought on ebay, from a company in Manchester (UK), for £14. Using It produces a mono recording on both tracks. In Final Cut Pro (Studio 3) the Audio Mixer from the Tools drop down menu has a stereo slider allowing the user to manipulate where the sound exists within the stereo spectrum.
If you're adding music in stereo - that you have recorded the voice in mono - will not show. But the improved sound quality will make your production sound far superior. The Apple EQ in FCP appears a little complicated. But have a play, as it thins the sound making it less muffled. Only the bottom few categories on the EQ filter are the actual EQ; the rest are just filter settings, such as Gain. I'm looking at this now!
There is one other diamond for both video and music production when you are trying to get by on a budget: Soundtrack Pro. With your video project selected on the FCP Audio track right click the mouse to bring up the sub menu. Select 'Send To' and the listed Soundtrack link for the Aiff audio project. A small menu will ask you to save the file to your folder. Just click on OK. Now your sound is in Soundtrack Pro.
Highlight the entire sound clip using either 'Select All' or left click the mouse in the top right-hand corner of the sound and drag it to cover everything. There are some folder options in the menu on the left. Select the Analyse folder. Now tick all of the box options on the left for noise pops, etc. Select 'Analyse'. The software will detect and audio problems and correct them, but you still may have a little white noise. This is where it gets clever. Select the entire sound again and set the Ambient Noise Point. This can be found in the Process drop down menu. Having set the noise point go back in and select 'Replace with Ambient Noise.' Ambient noise is a natural part of the sound spectrum. It's often in most audio mixes, but we don't notice. It's like giving the noise camouflage.
At this point you may be happy with the sound, as the only other matter to consider is the level on final cut pro. This also can reduce the white noise. If you are still unhappy there is one other EQ diamond on Soundtrack Pro: The Parametric EQ, which is found in the Effects/Single Band EQ. I suggest that you place all three of the sliders to the left and adjust the top slider until the sound is as required. The other two sliders will control sound spectrum differences reducing the noise to near zero without losing too much quality.
When you have finished adjusting the sound use the save option from the drop-down menu. It will ask you how you want to save the file. It's the top answer - just click on OK. This will automatically save the changes to the existing FCP sound file. You can continue with your video now. Warning: Always use slugs for gaps in audio and video. If you fail to do this your video will stutter when you add it to social media.
Add: Forgot to mention the noise reduction in Soundtrack Pro. Set the noise point then just click on OK for the noise reduction. Have a little listen to make sure that it hasn't added a nasty digital sound to your work. If it has move two steps backwards and avoid. Sometimes it works quite well, but it's dangerous!
There is an iPhone app to remove background noise from any video on your phone called Denoise. There is an app preview video which shows how it works. Technically it is all the same algorithm to measure background noise profile and suppress noise frequencies after that (two phases approach). Some more information on the app could be found here => https://blog.2gzr.com/clean-sound-in-your-videos-without-external-microphone/
Disclaimer: I'm one of the app developers and would love to hear your feedback
While removing white noise without tampering with the recording is a difficult task, a gate and tweaking of a compressor gets a descent result. of course the best way to achieve a perfect audio sample would be to record the audio again, but since its not your own recording, the gate, a compressor and an 8 band or higher equaliser help.