I have a 1996 mongoose bike and I need to replace the front rim 20x1.95 All I can find is 20x1.75 Can I use a 1.95 tire on a 20x1.75 rim
The answer is yes, a slightly narrower tyre will be fine.
If you read the metric numbers off the tyre you're safer, because those have a fixed meaning. Inch sizes depend on the exact context (a 27" road tyre will be too big for a 29" mountain bike rim, for example). You probably have a 45-406 tyre, and are looking at a 35-406 tyre. The "406" part says it will fit the rim, and because the "35" width is smaller than the existing ""45" width, it will fit into the bike frame. Since it's not much smaller, it won't be too small for the width of the rim.
Sadly we don't have a good way of linking all the various tyre sizing questions into one definitive answer that will come up when people with your problem search for it. The definitive answer exists - Sheldon Brown wrote it years ago. If you want to read more, or much more, that's the place to look. We have lots of similar questions, like this.
Edit in response to answer above: you're probably better off buying a new wheel, or a second hand wheel if there's a tip shop nearby. Buying a rim and paying someone to put new spokes between your existing hub and the new rim is almost certainly going to be more expensive than a new wheel.
Shipping on the wheel will be expensive compared to the cost of the wheel, so if you can get one locally you'll save more. You can get new ones on ebay for under $50, but a genuine mongoose one second hand might cost a similar amount.
I suggest taking it into a local bike shop and getting their opinionbefore spending any money, if that's an option.