What exactly does "nights" means when you book? Is it number of night you are going to sleep, or simply days you are going to stay?

Another way to think about it:

Endpoints are a common source of confusion. “Day” can mean “24-hour block” or “calendar days”. A stay that lasts exactly 7 times 24 hours will span 8 calendar days, except if you make it start exactly at midnight on a given day.

That's why it's much more convenient to think about “nights”. Usually a night goes from 3 or 4 PM on one day to 10 or 11 AM on the next. That's what you are booking if you are talking about accommodation and not a full 24 hours of stay. If you want more than that (i.e. arriving early or leaving late), you might have to pay more or risk having to wait before getting to your room.

To avoid any misunderstanding, you can simply communicate that you will arrive in the afternoon on a given day and leave in the morning on another one (i.e. you are not booking the night of Monday the 24th, you are booking the night from Monday the 24th to Tuesday the 25th).

Note that packaged tours or visa forms will sometimes also mention days and usually that would include both the day your travel is starting and the day it ends, so one more than the number of nights.

Usually hotels charge per night, which tends to be measured by you checking in after a certain time on the day before your first night, and checking out before a certain time on the day after your last night. There are sometimes discounts for staying only part of a day (a "day rate"). And there are sometimes extra charges for early check-in or late check-out (or sometimes these are waived - let them know as soon as you know your plans, or when your plans change).

In your example, 10am would tend to be an early check-in. In my experience with London hotels, the room likely might not be ready until the afternoon, but they have had places to store bags in the meantime. 10am is usually early enough for a regular check-out, but it's always best to check. So I would tell them when you will be arriving and leaving, ask when their times are to check in after and check out by, and assuming 10am is an early check-in, ask can you arrive then and leave bags, and when should you expect to be able to occupy the room.

Policies may vary by place and by hotel, so it's best to check to avoid surprises.

On your last day at the hotel, you usually have to leave by 10/11 am (you need to confirm the exact time with the hotel manager as there is no standard rule). If you exceed that limit, it will count as another night at the hotel.