What is a dollar bill with red serial numbers called?

Red seals were mostly used on $2 and $5 bills rather than $1 bills. As the banner across the top indicates, such a bill is called a United States Note.

US Notes were issued directly by the Federal Government, alongside silver and gold certificates, and eventually Federal Reserve Notes issued by the central bank.

US Notes and Federal Reserve Notes are "unbacked" notes; that is, they're accepted for payment because the vast majority of citizens are willing to accept them as representing a specific amount of money due to the stability, faith, and credit of the US Treasury. This contrasts with gold and silver certificates that were issued only in amounts that represented specific quantities of those metals on deposit with the government.

The US went off the gold standard during the Great Depression and gold certificates were discontinued. The same thing happened with silver in the 1960s so silver certificates were also discontinued. At that point US Notes and FRN's were the only forms of currency issued and were effectively the same. The government then decided to combine all currency production and distribution under the central bank to eliminate duplication of effort. Because many more FRNs were in circulation than US Notes, the latter were discontinued in 1966.