A bond's initially set Principal Amount, Term to Maturity, Coupon Rate, and Price, allow us to calculate its Effective Interest Rate, which, compared with the prevailing Market Interest Rate for similar bonds, allows us to calculate the bond's Market Value at any point in time.
To explain in detail, let's define a few terms with a consistent example (the above paragraph + the final paragraph is the quick answer to your question).
This is the stated 'receipt' value of the bond. If you hold a bond with a principal amount (also called 'face value', 'par value', or 'nominal value') of $1,000 (a common amount), then on the maturity date of the bond, you will receive $1,000 from whoever issued the bond.
Term to Maturity
Also called the Duration, this shows how long the bond will be outstanding for. A bond might have a 40 year Term, or a 1 year Term, depending on what the issuer needs the cash for. A long Term could be considered riskier if it means there is a higher chance of the issuer going bankrupt in the meantime, and it also might provide stability in interest rates that is attractive to someone who wants to lock in their investments (like a large institutional investor who needs to guarantee their own capital needs 40 years from now).
The Coupon Rate states the interest payments that the bond issuer will pay the investor over time, based on the Principal value of the bond. So a bond with a principal value of $1,000 and a Coupon rate of 1% would give the investor the right to $100 in interest payments each year (often paid semi-annually). The Coupon Rate can be anything a company decides - it might be a bit above or below "standard market rate" of interest, for various reasons which I will show below.
When a company or government issues a bond, they can choose to issue it for any price they want, regardless of the ultimate Principal Value. For example, a company could issue a bond with a Principal Value of $1,000, at a price of only $975. What this means, is that you buy the bond for $975, and then on its maturity date (let's say in this case the maturity is 1 year from now), you get $1,000. This is a discount of $25, and 25 / 975 = 2.6%, which means that the initial cost of buying the bond would earn you back 2.6% over the course of a year.
Effective Interest Rate
When a bond is issued, the effective interest rate is based on a combination of the initial price, the coupon payments, and the final principal value. Using the example above, a 1 year, $1,000 bond with a price of $975, and a 1% coupon rate, would have a combined effective interest rate of 3.6% (2.6% based on having a lower price, calculated above, + 1% additional coupon rate).
Market Interest Rate.
The Market Interest Rate is the rate that is currently expected on a similar bond, based on prevailing local interest rates and the risks of the issuer. For example, if the US Fed Rate is 0.25%, and corporate bonds from very strong companies are typically priced at 3.35% higher than the Fed Rate, then such a bond would be expected to have an effective interest rate of 3.6%.
The Market Value of a bond is what someone would expect to pay, in order to get an identical bond with the same Principal Value and Coupon Rate. At the time the bond is issued, then typically the company will try to precisely match the Effective Interest Rate with the Market Interest Rate.
So if the Market Interest Rate for an identical bond would be 3.6%, then the company will adjust the new bond's price and coupon rate to get an effective rate of exactly 3.6%, meaning the initial Price would exactly match the initial Market Value. If the company sets the Price too high, or the Coupon Rate too low, then it is unlikely to get 'fully subscribed', meaning if they tried to issue 10,000 bonds instead at $980 each, then instead of selling out and raising $9,800,000, perhaps they are only able to issue half of what they wanted, and would only raise $4,900,000. Likewise the don't want to set the price too low, or the might miss out on potential value.
Often, a company will use an intermediary financial institution to take the risk of pricing the bond, and that institution will likely guarantee full subscription, meaning if it is mis-priced then they bear the burden of purchasing any excess overpriced bonds.
Consider that the Market Value of the bond will fluctuate over the course of its life, until maturity
After a bond is issued, the Principal Value, Term to Maturity, and Coupon Rate will be unchanged. But the market interest rate will fluctuate based on real-world conditions. Maybe the Fed rate rises to 1%, or the company sales perform poorly, and the company is now considered risky enough that it needs an extra 1% risk-premium to compensate investors for the risk that the company will not be able to repay the principal value due to the chance of future bankruptcy.
In either such case, the bond would no longer be worth the initial price. Why would someone pay $975 for your bond with an effective interest rate of 3.6%, when a similar company the next day issues the same bond for only $965 [$35 face-value discount / $965 + coupon rate of 1% = effective interest rate of 4.6%, or about 1% higher effective rate than the bond you just issued yesterday]? If an identical bond to yours is available in the market at a cheaper price, then of course, the Market Value of your bond will drop to match it.
So now to your question - How does Principal relate to Market Value of a bond?
The Principal Value is the amount that the holder will receive from the issuer at the end of the Term, and is locked-in when the bond is issued. Alongside the Coupon Rate and Price, the Principal Value helps to calculate the Effective Interest Rate of the bond, and comparing this to the current Market Interest Rate gives us the Market Value of the bond at that point in time. [If the bond is priced correctly, then on that day, the Market Value will be equal to the Price].
After the date of issuance, the expected Market Interest Rate for similar bonds will fluctuate, based on the risk of the company and prevailing local interest rates, and if the market would demand a lower Price for a similar bond in the future, then the Market Value of the old bond will also lower. However, the Principal, Coupons, or Term will never change.