Engine Back fired, blew out muffler.

When a gas engine back fires and at times loses power at times just before back-firing it can sometimes suggest that the fuel tank's overflow tube, which is also responsible for allowing air to replace the fuel as it burns, otherwise, there would be tremendous suction by the time the tank was emptied. If you are losing power just before producing a back-fire event, the suggestion is, once the fuel reaches a certain level of suction, the engine dies because the suction is preventing the fuel from advancing, then, when the engine finally back-fires, it is because a blast of fuel was just pushed into the engine while it was coasting because of the shortage of air. If there is something you still don't understand, please respond accordingly.

the only time i have seen this is when the exhaust is blocked or weak. back fire would not normally do this. replace the muffler as you say. and check that the timing lines are in the right place. have you had engine management light come on at any time while driving. if time ok and engine has no lumps while driving then it would just be the muffler fault.

do you change your oil and what king of car you got

Usually this requires removing the timing cover, removing the timing gear, and moving the cam back into the correct position, Reassemble.

Check dist. cap and rotor.

No fire? There probably is fire, but out of time or some liquid (water, oil, PS fluid, etc.) got into the distributor cap and the rotor fire is shorting (sparking) to ground through the liquid.
Check spark anyway, pull a plug cable and set down on metal part of engine ground. Ahve someone crank starter while you look for spark from end of sparkplug cable to ground.
Let us know...

First check all wires for any thing loose.
The muffler blowing up leads me to think the engine is so far out of time the PCM cant correct it.

REally you need to get it to a good mechanic.

The clicking sound is from lack of oil or lack of oil pressure. When the lifters pump out then they have too much slack in them and thus the clicking or tapping sound. Sorry to say this, but the engine is either very tired or the oil pump has about had it.

check spark plugs. and igntion coil.

That sounds to me like the catalytic converter is going out. suggestion, take it to a muffler shop and have them check it out. As for the ticking and missfiring, mine has done the same thing for a while and I keep throwing out a vacuum leak as well as Throttle position control, cruise won't work as cruise, it will actually slow down on hills from 70-40 mph, but I know both are the problem, solution, the throttle position control is under the air intake, take the air box off of the throttle body and replace it after having autozone run a check. From there..... find someone good at mufflers as well as vacuum leaks and fix it. lol.