Constant Power to Garbage Disposal
First the bad news. If smoke came from the disposer the windings are burnt and it's never going to work. Unless when you connected the power cord (the tail), it was made up incorrectly. When you "touched the wire"  most likely some of the fine strands were loose and caused a "pop". 

So first take the disposer out from under the sink and check the connections and determine if the power cord was installed properly (you should be able to see the burns if it's at the connection point).  Then verify that you haven't burned up the windings (they should smell burnt). If there is any hint that the windings are burned return and replace the disposer.

Then check the breaker and make sure the power is on. Then use your tester to check the outlets under the sink. The outlet below the sink is usually a duplex and one side should be continually hot and the other should be on the switch above the counter by the sink. If you don't get power there start backtracking the circuit find the damaged location and of course repair the part.

If everything seems to be ok. Reinstall the disposer. 

I know you probably would like to know another way to check the windings on the disposal. Unfortunately I can't recommend a way to check it safely without a workbench or a good understanding of how to check the windings with an ohmmeter.
If you really want to know that you need to take it to a motor specialist or the manufacturer's recommended warranty shop.

Stay safe and good luck.