What you are most probably describing is a superficial thrombophlebitis of the penile veins (called MONDOR's Disease).
This is an inflammatory reaction to a clot within the vein, leading to a blockage and dialatation of the vein. WHile this is generally painless, it can also be painful sometimes.
Diagnosis is via DOPPLER ultrasound which shows blockage of the vessels.
Treatment is with anti-inflammatories (i.e. aspirin or ibuprofen) and
most resolve within 4-6 weeks.
Rarely in severe cases surgery may be
necessary.
An opinion of a urologist maybe advisable to confirm the same and for followup,
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Regards
Dr. Gupta