If you do it online, print a copy of the receipt just in case. If it's done online, the Police can check with the DVLA and it will show as paid and valid. There is, I believe, an option of printing off a temporary Tax Disc if you renew it online.
If you don't renew your tax disc and you are the registered keeper, you will recieve a computer generated warning letter 2 weeks later. If you still don't tax it, a computer generated fine is sent out to you on the 1st of the next month (exactly one calendar month after your tax expired).
If you use your car on the road in the interim you could be caught by the DVLA's mobile enforcement unit who can clamp and impound your car. The police and authorised traffic wardens can issue fines too.
The DVLA just want all the tax due for your vehicle so if you renew your tax inside of the month following your last discs expiry date they will happily backdate it and no more questions will be asked.
Hope that helps