[verb て-form] + いらっしゃる [verb て-form] + いらっしゃいます お + [verb ます-stem] + です / でいらっしゃいます
ていらっしゃる is the honorific form of ている: the て-form of a verb plus いらっしゃる, the honorific of いる. Use it for what someone you look up to is doing – a teacher, a customer, your boss, someone else's family. In polite speech it is いらっしゃいます.
An honorific raises the other person, so ていらっしゃる never describes you or your own side. For yourself, use ています, or the humble ております.
Like ている, it covers a lasting state, not only an action under way. けっこん結婚していらっしゃる is "is married", not "is getting married".
These go beyond the basic form.
お plus the ます-stem plus です – おまち待ちです, おでかけお出かけです – is a shorter honorific for an ongoing state, and お〜でいらっしゃいます a more formal one. Making both halves honorific, as in おまちになってお待ちになっていらっしゃいます, is correct but heavy.
In speech the い often drops: よん読んでらっしゃる, まっ待ってらっしゃいます.
いらっしゃる is also the honorific of 来る and 行く, so ていらっしゃる can stand for てくる or ていく as well. The everyday いってらっしゃい, "see you later", is 行っていらっしゃい.