1."I have lost lost my keys" or "I've lost my keys" is present perfect. This tense always tells us something about the situation now. If you say "I've lost my keys", this means that you don't know where your keys are now. You would say this only if you have not found them yet.

2."I lost my keys" is past simple. This tense tells us about an event in the past and does not necessarily tell us anything about the present. Without more context, it is impossible to know whether you have found them or not. To describe the situation in #1 above, many American English speakers would say "I lost my keys" instead of the more correct "I've lost my keys". In that case, you would have to infer from the context whether or not the keys have been found.

Here's another example:

Compare:

"I've broken my arm" = My arm is in plaster now (present perfect = information about the present)

"I broke my arm in January" = This tells us what happened and when it happened. We know nothing about the state of your arm now. (past simple = information about the past).

In the US, if you say “I’ve lost my keys many times.” or “I’ve lost my keys before” that would not mean that your keys are currently lost.

If you wanted to unambiguously describe not having your keys, you would have to say “I don’t have my keys.” or “I can’t find my keys.”

“I’ve lost my keys” (some time in the past)

“I lost my keys when I was taken to the hospital.” (In the past)

“I’ve lost my keys” is a perfectly natural thing to say in the US, but “I’ve broken my arm” less so. The clearer “My arm is broken.” or “I broke my arm.” (with a little context) would be more common choices. When you lose your keys it’s natural to not know exactly when you lost them. But breaking your arm is a well-defined event in the past.

Also, I disagree with the idea that it's merely a regional difference. In most cases, an American using present perfect instead of the simple past would do so to convey a slightly different meaning.

Mom, I'm in the hospital. I broke my arm. (telling a story about me and what I have done.)

I've broken my arm and I won't be able to help you move this weekend. (This shifts the focus away from the event of my arm breaking, to focus on the result. I don't think it sounds particularly British. You could also say "My arm is broken" or "I have a broken arm")

I'm a native US speaker. I feel that these two phrases have practically no difference in meaning in themselves.

Some people who have answered this question say that there are subtle differences. I think that a speaker who needs to express those differences clearly would need add other words and provide more context.

I think I would be most likely to say "I've lost my keys." I grew up near New York. As a reality check, I asked my wife, who grew up in the Midwest, what she would say. She said, immediately, "I lost my keys." I then asked her "can you think of any difference in meaning between I lost my keys and I have lost my keys?" And she said, immediately, "No." Then she added "Maybe I have lost my keys sounds like it happened less recently than I lost my keys."

This is one of those areas where people say confidently and definitely that there is a difference in US and British usage, and I just don't think it is all that clear.

I lost my keys- I use this when, right now, I need my keys and I don't know where they are. It is the simple past. Yesterday, I lost my keys. This morning, I lost my keys. I lost my keys immediately after receiving them.

I lost my keys somewhere in my apartment, I know they are here.

I have lost my keys has more possible meanings. Present perfect is not now, it is before now, leading up until this moment. The action, to lose, occurred before this moment (otherwise, you would not know it occurred) and could occur anytime until the present.

I have a recurring problem of losing my keys. I have lost my keys so many times it makes my head spin. (it drives me crazy.)

I want to go somewhere, but I have lost my keys this week. (I don't know when.) You could use lost here, but advanced speakers use helping verbs (have, has) with the participle (lost) for non-defined moments in time.