What a star note is
A star note is a replacement. The Bureau of Engraving and Printing prints it when a sheet is spoiled, and the serial ends in a star instead of a letter.
A replacement for a spoiled sheet
United States paper money is printed in sheets. When a sheet is spoiled during printing, inspection, or numbering, the Bureau of Engraving and Printing does not send that sheet out. It prints a replacement sheet for the same denomination and series.
The replacement uses a star at the end of the serial instead of the usual suffix letter. That is the mark collectors mean by a star note. The star is not a grade, a signature, or a claim about what the note is worth.
Why the run is smaller
Regular notes of a series are numbered in large blocks. Star notes are numbered in their own block, only large enough to replace the sheets that failed. A star run is often much smaller than the regular run for the same series and district, which is why collectors separate them.
Smaller does not mean scarce in every case. Some modern Federal Reserve star runs still contain millions of notes. An older series, or a run of a few hundred thousand, is a different collecting story from a recent run that was printed in volume.
The passport keeps the star, the series, and the district together so the run can be looked up later. A star with no matching row in the monthly tables is still a replacement. The missing row means the tables do not cover that printing, not that the star is decorative.
Any U.S. paper type can have a star
The star is a numbering practice, not a seal color. Federal Reserve Notes use it, and so did Silver Certificates, United States Notes, and the wartime issues. A blue seal or a red seal can end in a star. The seal tells you who issued the note. The star tells you it replaced another sheet.
On a modern note the star sits where the final letter would have been, after eight digits. The shape is a prefix, eight digits, and a star, as in B12345678★. Older typefaces used a star the same way.
The star is not a price
A star makes a note worth logging. It does not set a dollar value. Condition, series, denomination, and how many were printed in that run all sit beside the star. Two uncirculated star notes from different runs are not the same thing to a collector.
Rarity score is based on run size. Note that final value depends on multiple factors beyond rarity alone. The score used on this site comes from the size of the print run, which is a separate question from whether the note is a star at all.