Depending on the frequencies of the background noise, one might, or might not, have much luck removing it, without also destroying the human speech.
I use Adobe Audition to restore my Audio, and it is wonderfully effective, but one must be willing to spend quite a bit of time working on the piece. It allows for Noise Removal, after one has tested a section without the speech in it. It offers a lot of control as to the frequency ranges and the profile of the noise. After that Effect, one normally will need to work with the Parametric EQ, and possibly both the Low-Pass and High-Pass filters.
Now, Audition is not inexpensive and is a complete pro audio-editing program - with that power comes complexity.
However, some years ago, I found a much less expensive program, that does a surprisingly good job of doing just what you need - Magix Audio Cleaning Lab. I was blown away by how well this inexpensive program handled audio restoration. After having spent a full day with Audition, I was ready to just give up on a piece of audio. I had gotten ACL free in some Magix bundle, but had never used it. I gave it a go on the original piece, and was astounded by how good the result was. With some tweaks and then a few small Effects additions in Audition, I had an almost perfect piece of audio.
You might want to take a look at it.
And, just clarify, the De-Noiser is designed to reduce tape "hiss". Not to automatically remove background sounds (which is almost impossible to do).
As Hunt says, your best bets are the High-Pass, Low-Pass and Notch filters, which can be set to eliminate specific frequencies.
But don't expect miracles. There's really no magic way to get rid of bad sound and keep all of the good unfortunately.
Here is the low-down on the Denoiser Effect, which works exactly as Steve mentions:
The DeNoiser effect automatically detects tape noise and removes it. Use this effect to remove noise from analog audio recordings, such as magnetic tape recordings.
- Noise Floor
- Specifies the level (in decibels) of the noise floor as the clip plays.
- Stops the noise floor estimation at the current value. Use this control to locate noise that drops in and out of a clip.
- Specifies the amount of noise to remove within a range of ‑20 to 0 dB.
- Sets an offset value between the automatically detected noise floor and the value that you specify. This is limited to a range between ‑10 and +10 dB. Offset allows additional control when the automatic denoising is not sufficient.