I feel like I have a fish flopping around in my chest. Does this sound like a heart problem and if so, what?

Arrhythmia. Your heart is probably beating irregularly and/or harder than normal. It may be more annoying than an actual problem especially if you are young and otherwise healthy. You need an EKG to see exactly what type of arrhythmia is occurring. The treatment, if any, depends on the type of rhythm problem.
Palpitation Sx. Palpitations are an awareness of extra heartbeats which most of us have. The sensation can be single or multiple and can sometimes last minutes. Different people may feel other symptoms along with them such as dizzy, lightheaded or short of breath. The palpitations are almost always atrial or ventricular premature beats. Usually benign but they are also found with abnormal hearts.