Well it all depends on the size of the power plant!
Conventional stem plants (boiler + steam turbine) can have dual fuel burners in their boilers.
I used to be involved with small (1 to 5MW) gas turbines. They were usually natural gas fuelled, but in UK were usually specified as diesel / natural gas fuel. They would change over under load, and were often started on diesel (to excercise the pumps etc) then change over to gas. This was because gas contracts can be interuptible at the whim of the gas supplier.
For reciprocating engines, the gas fuelled engine came first, invented by Otto (often forgotten - Dr Diesel came later).
Nowadays as catserveng says, we have several variations:
Direct Injection Diesel Engine - liquid fuel only.
Larger (5+ MW) engines can be configured to run on Heavy Fuel Oil (HFO. They usually start / stop on standard diesel fuel to flush out the HFO, which can be solid at 35C. They could run on straigt diesel, but this often costs twice as much as HFO.
Dual fuel engine - This basically a diesel engine, with fuel gas admitted into the inlet manifold, uasing about 10% of the engergy as diesel fuel as a "pilot" to ignite the gas. Not used much in smaller engines now, I think, because of the cost of the diesel fuel, and I understand that they are difficult to get to meet modern exhaust emission limits. Typical application used to be in a sewage works, where the engine was fueled with the methane from the sewage digester. The advantage being, if the gas was not available, the engine could continue to run as a straight diesel engine.
Spark ignition gas engine. This is basically musch like a car's pertrol engine. With a gas mixer in place of the carburettor. Advnmtage: very efficient, relatively low emissions. They can have a gas fuel changeover system (again at a sewage works), but these are fairly uncommon in UK, at least. Modern gas engines are quite hi-tech and have computer management systems - very much like the modern car engines.
Finally - there is what I call a "bi - fuel" engine, which is a standard diesel engine with an add - on kit to enable it to run on gas as well. These seem to take higher levels of liquid fuel as a pilot fuel - about 20% say. Again, may have difficulty in meeting emissions limits and efficiency may be poor.
So as we say, its horses for courses!
Its possible to convert an engine from one type to another (say with the bi-fuel kit) but otherwise it is a major rebuild (pistons, liners, cylinder heads etc etc).