Where, the conversion factor for a bond (by John C. Hull) is set equal to the quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per annum (with semiannual compounding)
The purpose of conversion factor is to make bonds in the delivery basket more equally deliverable (theoretically anyways, but the process is not perfect).
This is an important design decision for bond futures. Without conversion factor, bonds with low coupon rates will have significantly lower prices than their high coupon peers in the delivery basket, making them de facto cheapest-to-deliver (CTD) issues. A smart investor can buy up all available supply of these low coupon bonds, making it impossible for other market participants to make delivery ("delivery squeeze").
With conversion factors, the bonds become much more similar from a delivery price perspective. If the CTD is cornered by one investor and not obtainable, we simply move to the next possible CTD without incurring too significant a penalty.
In this chart below, I'm drawing the delivery prices of three bonds under various parallel yield shifts, WITHOUT the conversion factor adjustment:
Clearly the 4% coupon issue is dominantly cheap. If that issue is not available, I'd have to deliver the next cheapest bond (the 6% issue), whose price is much much higher.
The next chart shows delivery prices for the same three bonds WITH conversion factor adjustments:
As said, the process is not perfect, but the differences are now much more tolerable.