What is roof underlayment made of?

Asphalt felt

A base sheet of paper fibers or fiberglass is soaked in asphalt. The two classic weights are #15 and #30, with #30 thicker and tougher. Products made to ASTM D226 or D4869 meet code. Felt is breathable and inexpensive, but it tears easily, wrinkles when it absorbs water and dries out over time.

Synthetic

Polypropylene or polyethylene is woven or spun into a strong fabric and coated to shed water and give a slip-resistant walking surface. It is much stronger than felt, does not wrinkle and resists UV longer. Synthetics are tested to their own standards and usually carry a code evaluation report or listing, so check the label. Most act as vapor retarders, though some are made breathable.

Self-adhered membrane

Asphalt modified with SBS rubber, reinforced with fiberglass or polyester, topped with a polyethylene film or granules, and backed with adhesive and a release sheet. Made to ASTM D1970, it bonds to the deck and seals around nails, which is why it goes in valleys, around penetrations and on low slopes. High temperature versions go under metal roofs, where the panels get hot enough to soften standard membrane adhesive.

What a Georgia roof usually gets

Synthetic over the field, membrane in the water-prone spots, and high-temp membrane under metal. See synthetic underlayment vs felt and ice and water shield in Georgia.