For example, suppose that I'm going to travel London and will arrive at 10am, 1st December. And will stay a week, means will leave 10am, 8th December. Technically it's 7 nights, but also 8 days. Should I book 7 counts(nights) or 8 counts?
Accommodation bookings count the nights you spend simply because that's what you need accommodation for: to sleep. This is based on the somewhat valid assumption that your main night time activity is sleeping.
I understand where the confusion might arise from between day and night counts. Because nights are interposed between days you think you are staying for 1 day more. However consider that usually check-in time is in the afternoon and check-out is in the morning so effectively you are occupying the accommodation for the exact numbers of hours (computed as
nights * 24h) you booked it for. To this purpose consider a realistic 3 night booking with check-in at 3pm and checkout at 11am. Total booking time is
2 * 24 + 20 = 68h which is
~ 3 days.
Your 7-Night Reservation
You say you plan on arriving on the 1st December and leaving on the 8th of December. That equates to a booking of 7 nights. Since you mention an arrival time of 10am, the actual check-in time depends on the hotel/hostel. Most of these usually check guests in in the afternoon, to leave time between check-outs to clean the rooms. Check-out times tend to be before noon, for the same exact reason. To be sure, ask the hotel for specific check-in and check-out times.
However, if you arrive before your room is ready you can most definitely ask for them to keep your luggage whilst you go to the nearest pub for a good old English breakfast. This should be a free service.
As mentioned in the comments below, if you desperately need the room for 10am because you want to shower/sleep then you might have to book for an extra night (the one between the 30th November and 1st December). If you do so make sure you tell the hotel you will be arriving on the 1st rather than on the 30th, to ensure they don't consider it a no-show and cancel your booking (most probably after charging you for the whole amount nonetheless).
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.