It is when we swear or take an oath on other than Allaah.
Swearing on ANYTHING, OR ANYONE other than Allaah, is an extremely serious matter and is considered SHIRK, even if the person swearing is telling the truth.

The Prophet (pbuh) said:

"Whoever swears by anything other than Allaah is guilty of kufr or shirk." (Saheeh, narrated by Ahmad, Abu Dawood, al-Tirmidhi).

Yet, it is very common in our daily lives that we see and hear people swearing by a vast multitude of things. Some people swear by the Prophet (pbuh), Ali (RA), AbdulQadir Jilani, or some other righteous person, or even the Ka'aba or the angels.Sometimes they swear by their honesty or honor, such as in the Scouts' promise they say: "I promise by my honor..."

Also we see people, especially from the Indo-Pak subcontinent swearing by "the life of their child" or "their aged mother" or "their dead mother's grave" or their "sister's honor", etc. (being greatly influenced by the Indian Film Industry)!

The the Prophet (pbuh) said:

"Whoever swears by honesty is not one of us." (Abu Dawud-Saheeh)

So we have to really beware of this serious issue and not swear by anything except Allaah alone, or by one of His attributes. and obviously only when it is the TRUTH.

Actually, it is recommended in the Sharee'ah to be cautious about swearing oaths, and not to swear except when essential.

"And protect your oaths (i.e. do not swear much)"[al-Maa'idah:89]

And if someone LIES when he is swearing, then this is even worse and a major sin.

The Prophet (PBUH) said, "Of the major sins are: Associating anything in worship with Allah, disobedience to the parents, killing without justification and taking a false oath (intentionally)."[Al-Bukhari].