no default constructor exists for class

If you define a class without any constructor, the compiler will synthesize a constructor for you (and that will be a default constructor -- i.e., one that doesn't require any arguments). If, however, you do define a constructor, (even if it does take one or more arguments) the compiler will not synthesize a constructor for you -- at that point, you've taken responsibility for constructing objects of that class, so the compiler "steps back", so to speak, and leaves that job to you.

You have two choices. You need to either provide a default constructor, or you need to supply the correct parameter when you define an object. For example, you could change your constructor to look something like:

Blowfish(BlowfishAlgorithm algorithm = CBC);

...so the ctor could be invoked without (explicitly) specifying an algorithm (in which case it would use CBC as the algorithm).

The other alternative would be to explicitly specify the algorithm when you define a Blowfish object:

class GameCryptography { 
    Blowfish blowfish_;
public:
    GameCryptography() : blowfish_(ECB) {}
    // ...
};

In C++ 11 (or later) you have one more option available. You can define your constructor that takes an argument, but then tell the compiler to generate the constructor it would have if you didn't define one:

class GameCryptography { 
public:

    // define our ctor that takes an argument
    GameCryptography(BlofishAlgorithm); 

    // Tell the compiler to do what it would have if we didn't define a ctor:
    GameCryptography() = default;
};

As a final note, I think it's worth mentioning that ECB, CBC, CFB, etc., are modes of operation, not really encryption algorithms themselves. Calling them algorithms won't bother the compiler, but is unreasonably likely to cause a problem for others reading the code.

You declared the constructor blowfish as this:

Blowfish(BlowfishAlgorithm algorithm);

So this line cannot exist (without further initialization later):

Blowfish _blowfish;

since you passed no parameter. It does not understand how to handle a parameter-less declaration of object "BlowFish" - you need to create another constructor for that.

Because you have this:

Blowfish(BlowfishAlgorithm algorithm);

It's not a default constructor. The default constructor is one which takes no parameters. i.e.

Blowfish();

A default constructor is a constructor that either has no parameters, or if it has parameters, all the parameters have default values.