Market value will be what Galss's Guide values the vehicle at the time of the incident (including adjustments for condition, mileage, etc).
This is based o na natinal average of selling prices, so it MAY not necessarily reflect what cars sell for in your area, but we usually also have a quick look on Autotrader.
The mindset of ''reject the first offer'' sadly remains, and this is no guarantee of an increase in value - research the valuation using the internet (ie Autotrader as previously mentioned), and gather adverts ready to support your case if applicable.
The adverts would need to show vehicles of the same make, model, age, number of doors, and the mileage should not be lower than your vehicle in order for the adverts to be classed as comparable.
I, personally, get very few rejections. We can't go offering stupid amounts as we are goverened by the ombudsman who would simply overturn our decision at our cost if the offer was not deemed to be fair.
You will always get people who argue their car is a classic (ie rusty fiesta/mini/nova), or simply are greedy - this will not work - be realistic and look at cars currently for sale to get a good idea of what you should get.