What is Shirk, and what are some examples of a person engaging in it?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

al-Shirk (polytheism) is of two kinds: major Shirk and minor Shirk. The major Shirk means to believe that Allah, The Great and Almighty has a partner: a god or Taghut (who exceeds his limits as a slave to Allah) who can manages the affairs of this universe.
The major Shirk also means to worship somebody/thing with Allah, i.e. to pray, slaughter or vow for some Taghut, grave and so on.
It also means to believe that similarity between Allah and His slaves in Attributes and Names or to, for example, deny some of the Divine Attributes and Names.
In fact, one who commits such form of al-Shirk becomes non-Muslim, i.e. brings oneself out of the fold of Muslims.
As for the minor Shirk, it is also prohibited by Sharia. Since it may lead to the major one. Examples of this Shirk include: Taking an oath by other than Allah and al-Riyaa (duplicity and performing acts for the sake of showing off and others' praise). This form of Shirk does not bring one out of the fold of Islam and Muslims, but it is a great sin.
Allah knows best.