The conversion price is the price (ie the amount of the investment capital) that will be exchanged for each share issued to the investor at the time of conversion. The time of conversion will depend on which event occurs first of the successful completion of a new funding round, an exit (eg a share sale), an insolvency event, or the longstop date.
The conversion price when conversion is part of a funding round should generally be at a discount to the usual price per share, to compensate the investor for the risk they have taken by transferring funds in advance and by investing early. The conversion price discount is usually between 10% and 30%, with 20% being the most common discount.