Many of the problems in the UK came about because banks had loaned money to people who couldn't afford to repay it. (They largely weren't direct loans to customers; they were packages of mortgage debts bought from the USA and elsewhere, where the original lenders had been over-optimistic about the ability of house-owners to repay their mortgages).
If British High Street banks would have been allowed to go bust, there would have been massive economic problems here (with customers losing their savings and thousands of businesses having to close when the banks' receivers demanded immediate repayment of overdrafts). So the Government stepped in to help.
However national governments need to get the money they spend (both on regular spending and upon supporting their banks) from somewhere. That has traditionally been through taxation but, increasingly over the past half century, governments have relied on borrowing money (by selling interest-bearing bonds). That plunges actual countries (rather than just their financial institutions) into debt.
The UK's debt has been rising at a rate which is clearly impossible to sustain in the long term. It continues to rise by £5000 every second[i, which is why all political parties agree that cuts in Government spending are needed. (They only disagree on the speed at which such cuts should be introduced).
Cyprus and Greece both had somewhat similar problems with their banks but their governments were [i]already] far more in debt (relative to what was taken in through taxation) than the UK. They tried to borrow more money to see them through the crisis but (because those countries were seen as risky investments) they had to offer far more interest on the bonds they were selling than countries like the UK and Germany did, which increased the rate at which they were falling into debt.
The position was eventually reached where nearly all of the money taken in taxes was being spent on paying back loans, with no money left to run the country (e.g. by paying public sector workers). The EU agreed to help out but only on condition that those countries immediately started 'to put their houses in order'.
Because the UK isn't in the Eurozone, there will be no bail-outs if a similar situation arises here (which it inevitably will if the Government - of whichever political colour - can't sort out the mess that we're in). If we don't get it right, we're headed for far more problems than any Eurozone country has had to face!