I'm trying to understand how junk bonds would incite investors to purchase them, did they offer higher coupon rates then the prevailing market or were they simple priced cheaply in order to ensure a higher yield?
Not all bonds are issued as junk. Some may start out investment grade (and thus have a relatively low coupon) and turn into junk meaning that their price would drop due to an increase in the default risk of the issuer.
Generally bonds are issued close to par, so bonds with a higher risk of default would need to pay a higher coupon in order to compensate for that risk and get a price near par.
This is one of the reason that yield is such an important measure for bonds. It is a way to compare bonds that have different coupons and prices in a meaningful way.
Note that bonds are classified as "junk" because they have a high risk of default. The price is a reflection of both the coupon amount and the risk of default, so a bond might have a low price OR a high coupon and still not be classified as "junk".
Junk bonds refer to higher risk higher yielding fixed income securities. As you mention there are multiple ways for a junk bond to become junk. A common definition of what a junk bond is comes from the ratings agencies, for example S&P considers anything BB or lower a junk bond.
So, either you begin at BB or lower, in which case you will need to offer investors a higher yield to take the credit risk and invest in your bonds.
Or, you start at a higher rating and pay a lower yield, and then for some reason your credit begins to decline, and eventually you move into junk territory.
In either case, your YTM (yield to maturity) will be high, and this is comprised of both the coupon and any discount/premium the bond is trading at.