The Q Gospel is a potential source document for various books of the new testament, containing a collection of unmitigated sayings and teachings of Jesus Christ.
Its existence is alleged by various scholars of Christian Theology including B. H. Streeter, based on similarites between New Testament gospels such as Matthew and Luke, which contain minor variations on many quotes supposed to have been said by Jesus and therefore are alleged to have used as their source the same now-missing document.
Just one of numerous examples would be Jesus's 'discourse on judgmentalism':
"Judge not, that ye be not judged. For with what judgment ye judge, ye shall be judged: and with what measure ye mete, it shall be measured to you again."
"Judge not, and ye shall not be judged: condemn not, and ye shall not be condemned: forgive, and ye shall be forgiven"
Like students writing essays based on the same source material, Matthew and Luke have written these teachings with slight variations after taking them from a primary source.
The other explanation is that these books i.e. Matthew and Luke, copied one from from the other, as it is generally considered to be the case that the Gospel of Mark was also used as a source by these two later-written gospels. However, by cross-sourcing quotes and references from these two books, biblical scholars tend to agree that neither Matthew or Luke is dependant on the other in the way that they are both dependant on Mark.
It is highly unlikely that any written version of the original Q Gospel will be unearthed, especially since, as a set of teachings, it may have existed mostly or solely as part of the oral tradition, passed down by followers only verbally as many religious texts continue to be in parts of the world even to this day. This would further explain the discrepencies between the quotes in different gospels.
However, finds of this magnitude can occur, even centuries or millennia after the fact. The most notable example being the 972 "dead sea scrolls", found stored in jars in a cave in Gaza in the 1940s, which included the earliest known written versions of many biblical texts as well as many (30% of the identified texts) previously unknown extra-biblical writings which cross-referenced or confirmed certain facts and dates in the canonical Bible.
Many of the dead sea scrolls were found in very poor condition and some have still yet to be identified. 
A Selection of New Testament Stories Alleged to Have Originated from the Q Gospel
- The Beatitudes
- Love your enemies
- Golden Rule
- Judge not, lest ye be judged
- The Test of a Good Person
- The Parable of the Wise and the Foolish Builders
- The Parable of the Lost Sheep
- The Parable of the Wedding Feast
- The Parable of the Talents
- The Parable of the Leaven
- The Parable of the blind leading the blind
- The Lord's Prayer
- Expounding of the Law
- The Birds of Heaven and The Lilies in the Field