Use intense eye contact to captivate someone.

Pretend that your eyes are glued to your conversation partner.

Maintain eye contact even when he or she is done speaking.

When you must look away, do it very slowly.
Imagine your gaze as sticky candy, stretching until it finally breaks.

Be less sticky in a man-to-man conversation, but do increase eye contact slightly more than normal

If you want to hold intense eye contact, there must be at least two other people - your target and one other person.
Don't try this on strangers in public places as this packs a powerful punch.

Watch your target person even if someone else is talking.
No matter who is speaking, maintain eye contact with the person you want to impact.

For a gentler effect, you can take your eyes off from the listener to the speaker but bounce back to the target when the speaker finishes a point.
Your target will still think that you are interested in their reactions.