Transfinite ordinals are a type of number used in set theory to describe the sizes and arrangements of infinite sets. They extend the concept of ordinal numbers, which represent the position of elements in a well-ordered set, beyond finite limits. The first transfinite ordinal is denoted as ω, representing the order type of the natural numbers. These ordinals can be used to compare different infinite sets and understand their structure. For example, ω + 1 represents the order type of the natural numbers followed by one additional element. Transfinite ordinals help mathematicians explore the properties of infinity in a rigorous way.