Battery started smoking during charging. Smoke was puffing out from time time. Can these battery explode?

If the battery is very dead, it can take a while for the dead battery to regain enough charge to start. If you are jump starting from a jump box and not a running vehicle, there may not be enough power available to get the charge to a level that can start the vehicle. Hook up the jumpers cable to a running vehicle and allow 10-20 minutes for the battery to gain some charge. Monitor both batteries carefully during this time and should any unusual amount of heat or if smoke starts to come from either battery, shut everything down and disconnect as quickly (and safely) as possible.

You have a major short in your radio or in the harness inside the dash do not run this vehicle until you solve this issue. It is a fire hazard and could possibly cause the battery to explode

Yes, it could be the starter. It also could be the battery. I've replaced the battery three times in my 20 year old car, but the original starter is still fine. I'd have the battery checked before I replace the starter.
In another car, I had symptoms like bad starter, but in that car, the starter was really expensive, and on a hunch, I had the charging system tested. The "new" battery was defective, saving me hundreds when I didn't need to replace the starter.

Also check cables and connectors. One connector could have become hot and made a shortage.

Sounds like you have a weak battery..try giving it a good charge and try again many times engines will not start with a battery that turns the engine over but not fast enough to start.. Good luck.

replace it, u have a shorted cell.

I am having problems seeing if you have any more data besides "smoke alarm keeps chirping." so I will ask; have you tried changing the battery?

battery gives off oxygen and hydrogen ,both hight inflamable,loose connection caught this gas alight,possibly a loose battery terminal.check it out you were lucky battery didnt explode

Take it back to someone else. SMOKE is bad. Electronics tend not to work when you let the smoke out of them. Even tho the operating system may say its charging or that it is running off of mains power it may not be getting current to the battery to charge properly because the some has been let out of the components in the battery.
If you have troubles returning it simply tell the manager on duty that this is a Lithium Ion battery that may now be DANGEROUS to you and your loved ones. And is considered a letheal device according to the TSA that looks for Batteries that have been compromised.
Do not expect much from ACER, sorry, used to work in tech support here and ACER has not been known for quick helps. but complain to the manager where you got it and he may take it back in order to avoid a lawsuit from an EXPLODING battery.

I have a First Alert Smoke Detector Model SA 86RAC.When I changed the Battery today it beeps.How do I fix this problem?