You can find excellent technical tutorials on the internet but here are some basics:
I agree with others that 8x is a good magnification. 10x is harder to hold and, particularly with small binoculars, harder to work with.
The second number, like 50 in 7x50 is the diameter of the objective lens which drives the size and cost of the binoculars. As @stepford mentions, dividing the numbers 50/7 gives you the exit pupil. Glasses with big exit pupils, like the 7x50, are sometimes referred to as "night glasses." But .... more is not necessarily better because once the exit pupil is larger than the pupil in your eye it's game over. No more benefit. This is important to us because as we age, our pupils lose ability to dilate. (
https://en.wikipedia.org/wiki/File:Eye_pupil_dilation_change_with_age.pdf) We see this as poorer night vision but it also means that we can no longer get any benefit from large bino exit pupils. OTOH, small exit pupils like from a 10x20 miniature bino make the bino very fussy to use. The tube spacing has to be very exact to get the best view.
Eye relief is another thing to be aware of, especially if you wear glasses. Related to this is adjustable eye cups which can be collapsed when the bino user is wearing glasses.
Finally, there are two flavors of binos: porroprism and roof prism. Porroprism binos have the familiar "crook" in the barrel; the eyepiece and objective lens are offset. Roof prism binos have the lenses apparently in line but there is some trickery going on inside those straight-looking barrels. Porro prisms, generally speaking, give higher quality images for the price because they are cheaper to make than roof prisms. Roof prism binos are more compact but more expensive for a given level if image quality.
Like many things, more money gets you higher quality images but the incremental returns diminish as the price rises.
Lens coatings are important but if you're buying new or near-new you'll be getting good coatings. It's only when you are looking at older binos like WWII military glasses that you're in the pre-modern era for coatings. Don't go there.
Make sure whatever you buy has center focus, not an adjustment ring on just one of the tubes. This latter option shows up on military glasses mostly.
What to buy? $150 doesn't really get very far towards bino quality. I would suggest lurking on your local CraigsList, comparing prices to eBay "Sold" prices for identical binos and checking bino test reports from the birder community. I would avoid the low-end manufacturers like Tasco, Bushnell, and Swift. Nikon makes a range of binos but again I'd avoid the low end. Many expensive brands have already been mentioned here. You will not find modern Zeiss or Swarovski glass in your price range and you don't need them anyway.
In our local CraigsList today there is a pair of Nikon Monarch 5-series binos at exactly your $150. That's about half of Nikon's new list price. Review here:
Nikon Monarch Binoculars 2017 Comparison Review I suggest you lurk your local CraigsList waiting for something like this.
I have what is optimum for me, Leupold Katmai 8x32s. (
https://www.amazon.com/dp/B001HN5GPI) Unfortunately they are out of production and hard to find. I had a pair stolen and lurked on eBay for most of a year before I found the pair that I have now. They are the most compact "real" binoculars that I know of.
My wife has a pair of miniature Zeiss 10x20s that I bought before I really understood binoculars. They provide a good image but are very fussy to use because of the small exit pupils and poor eye relief. Their only real claim to fame is their small size.
HTH
