A private number plate also known as a cherished plate is a number plate that you can assign to your vehicle where the owner can pick what numbers or letters they wish to use. These are normally associated with a name / nickname / date of birth or company name and are growing in popularity. There are very strict rules that apply to private number plates concerning spacing, combination of numbers and letters you can have and even the word they are trying to suggest. There was a case recently when someone tried to purchase a plate that was blatantly made to look like swearing and the DVLA quickly banned it.
If you are no sure which one you want, there are plenty of companies out there that will find a plate that closely matches your request. You should not have to pay for this service, as many companies will find the plate for you for free although you may have to pay if you wish them to try and acquire it from another owner.