When you are about to lose an eye, you will be told that you will lose some depth perception. The key is "some" which for most people turns out to be "very little".
Oh, yes, it will become more difficult to do some things, such as thread a needle. It will also become more difficult to play some types of sports. But for the most part, the loss of an eye will probably not significantly affect your depth perception or your life.
How do you get depth perception in the first place? Because your eyes are a couple of inches apart, each of your eyes sees a slightly different image, shadows, etc., and your mind has learned to compare the two and make judgments on distance. Depending on your age, your mind has probably made millions of judgments and over the years has been trained to recognized distances.
Outside of about 20 feet, the couple of inches between your eyes doesn't make any substantial difference, and so in gauging distances outside of 20 feet it doesn't matter if you have one eye or two. A guy with one eyes sees the world outside of 20 feet the same as the guy with two eyes. It only really matters for stuff closer in, and the closer in the more significant the difference.
Again, you get depth perception because your two eyes are a couple of inches apart and see a slightly different image which your mind compares. If you only have one eye, you only get one image unless . . . [gasp] . . . you move your head slightly from side to side so that you get a couple of different images from your one eye for your mind to compare. This allows you to, essentially, with one eye mimick what you would see with two eyes, and thus alleviate the close-in depth perception problem.
Where this simple and effective trick does not work is with an activity which is fast-paced, such as sports. You will simply not have enough time to move your head around and get different images if, for instance, you have a tennis ball coming at you at 90 mph.
So, what can I tell you about sports? Not much, because yet I have not had enough practical experience playing many sports with just one eye. I can tell you the obvious, that some sports will not be effected at all by your one-eyedness. These will include jogging, swimming, water-skiing, snow-skiing, snowboarding, surfing, etc. Some sports, such as tennis might be significantly affected. Other sports may or may not be affected, depending on the position you play. For instance, I doubt that your one-eyedness would make much of a difference if you play the pitcher in baseball (the plate is outside of 20 feet), if you play in the post position in basketball, or if you play on the offensive line in football.
Remember that your brain will have to re-train itself to cope with your single vision. You can help it out by practicing your one-eyed depth perception, such as by laying on your back and repeatedly throwing a tennis ball straight up in the air and catching it as it comes down (be sure to wear safety goggles when you do this).
There are some things which might cause you trouble at first, which are solved if you just do them a little more slowly than you did before. An example is shaking somebody's hand -- don't feel uncomfortable slowly extending your hand and letting the other person grasp it (I haven't found this to be a problem at all). Steps can be weird at first, and of course you need to be careful of the blind spot, caused by your nose, on the side on which you lost your eye.
Other problems can be solved by just proceeding slowly and cautiously at first. An example is pouring orange juice. If you hold the pitcher a foot above the glass you may or may not hit it. What you should do is lower the pitcher (or raise the glass) until the lip of the glass almost touches the pitcher, and then pour so that it is a "sure thing". After a while, your mind will learn to compensate for your one-eyedness and such things as pouring will once again become second nature to you.
Driving, and particularly parking, may pose some depth perception problems. After my eye was removed, my girlfriend placed a board on the garage floor to indicate to me when to stop my car. However, I've learned that it is easier to simply find a familiar spot on the wall next to my door, and to learn to stop when I have lined up with it. Like everything else, parking with one eye will simply take some practice, but you can expect to get good at it over time.
Okay, folks, it's been more than six months since I lost my eye, and I can definitely tell you that:
I can lay on my back, throw a tennis ball 10 feet into the air, and catch it on the way back down, no problemo. I don't have any problems judging wave distance when I'm out surfing (well, at least no more problem than I had before). Depth perception is not an issue in parking (the loss of peripheral vision does occasionally cause some difficulty, but that's not depth perception).
Personally, I think the mind compensates very quickly for the loss of the other eye, to where after a while it is no longer noticeable (not that it was very noticeable in the first place).