You can now pay of Car Tax on-line, and if you print off the receipt that the DVLA send you electronically, you'll be able to drive your car until your nex Tax Disc arrives - but check with the Police beforehand.
My Tax ran out when I was overseas working during Januray, I sorted it out on-line & my car was able to be used stright away (DVLA informed me this was the case)
Police can generally grant you a 14-day 'grace' when it comes to you having paid for your Car Tax, but being unable to display the disc, and they'd normally provide you with a 'producer' which means you'll get 7 working days to produce the required documents at a Police Station of your choice.
Hope this helps!