The area you have described as the 'city center' with bars and clubs is typically referred to by Londoners as the "West End"....
The West End of London (more commonly referred to as simply the West
End) is an area of Central London containing many of the city's major
tourist attractions, shops, businesses, government buildings and
entertainment venues (including the commercial West End theatres).
The "West End" is an informal term and does not have strict boundaries.
Hotels will always have a postal code. If you are booking a hotel, you can examine the hotel's postal code to see if it is inside the "West End" or immediately adjacent to it. Postal codes start with the Post Code Area, which is the first one or two letters followed by a number, which signifies the district.
Combining the area and district will tell you where it is relative to the 'city center'.
In your case, a postal codes starting with WC or W1 (City of Westminster) would be directly inside the West End.
Examples are W1G (Marylebone) and W1F (Soho) and W1K (Mayfair). Covent Garden and the 'theatre district' is in WC2.
Lots of visitors like to book in the Paddington district and there are countless small B&B's and tourist hotels to cater to visitors.
Establishments in Paddington would have postal codes starting with W2. See http://en.wikipedia.org/wiki/Paddington
Areas immediately adjacent to the West End would have postal areas/districts like W2 (Paddington), SW1 (Belgravia), and to the north, NW1, NW8, and so on.
As you get further out, the district numbers will generally increase although not always. You'll need to refer to a map for districts greater than 1.
NOTE: the 'City of London' itself uses post codes starting with EC. This is the financial district and does not have the night life commonly associated with 'London'.
There is an interactive postal code map at http://www.free-postcode-maps.co.uk/postcode-maps/towns/london-postcode-map.php#.VHn9_jGsV8E
Also see Where to stay for London nightlife?