Learn Russian: Этот отдыхающий болен и не выйдет из номера. - This vacationer is sick and will not leave the room.

Отдыхающий is a substantivized participle. It comes from the present active participle of the verb отдыхать (to rest or vacation), so it literally means "the one who is vacationing." Even though it acts completely as a noun here, it retains its adjective-like endings and declines exactly like an adjective.

If the vacationer were a woman, how would this sentence change?

Because отдыхающий is a participle acting as a noun, it must change to reflect the natural gender of the person. For a female vacationer, the sentence becomes: Эта отдыхающая больна и не выйдет из номера. Notice that the demonstrative pronoun (Эта) and the short-form adjective (больна) also change to the feminine form.

Why is the short-form adjective болен used here instead of больной?

Short-form adjectives are strongly preferred when expressing a temporary state. Here, болен emphasizes that the vacationer is currently sick. If you used the long form больной, it could imply that he is a chronically ill person, or it might even be read as another substantivized word meaning "the patient".

Why does the sentence use номер for "room" instead of комната?