For an underage bride or groom to marry, a parent or guardian had to sign a piece of paper indicating the marriage was permissible.  The age of consent for males was usually 21 years, for females 18 years.  This signed slip was brought to the county clerk and had to be submitted before a marriage license could be obtained.  The existence of the slip was usually recorded in a county record book on the day the license was procured, though the date on the slip itself might have been several days earlier.

The loose permission slip itself was likely lost over time, but some counties still have them.  Other counties have turned them over to a county genealogical or historical society for safe keeping.  It is worth contacting the county clerk or county historical or genealogical societies to see if such loose records exist.  But don't expect them to do a blind search.  You should know the couple's names and at least approximate date of marriage before you make your request.

With communications not being what they are today, there was some need to insure that someone representing themselves as "marriageable" really was; that is, that they were not already married and not committing bigamy.  The marriage bond was a sizable amount of money for its day, which friends and/or relatives of both the bride and groom (sometimes the groom himself) posted guaranteeing that both members of the couple were indeed eligible to marry.  This bond was forfeited if such turned out later not to be the case, so the posting of the bond gave some measure of assurance to each member of the couple that neither was misrepresenting themselves to the other.

The signed bond was usually submitted to the county clerk the day the license was obtained, but the bond itself, which was a loose piece of paper, may have been signed days (rarely weeks) earlier.  Like the permission slips, some original marriage bonds may still be extant at the county clerk's office or with the county historical or genealogical society.  More often, they have not survived, and we know of them only because their existence was recorded in a county book of "Marriage Bonds."  But the book itself was not the bond, the piece of paper was the bond — just as the wills in a county will book are recorded copies, not the original wills. 

Here is an example of a Marriage Bond (for Thomas Thompson and Delilah Davis of Christian Co., KY, in 1815). 

The marriage license was issued by the county clerk upon receipt of the signed marriage bond (and permission slip, if needed).  The date the license was obtained is the first of the two dates occurring in most 19th-Century county marriage books.   The license was then issued (another loose piece of paper that may or may not have survived) granting permission for anyone so qualified, that is, an "MG" (minister of the gospel) or "JP" (justice of the peace), to perform the ceremony.  The ceremony may have taken place that day or days (rarely weeks) later — and possibly in another county. 

Here is an example of a Marriage License (again, for Thomas Thompson and Delilah Davis), on the back of which is the minister's return. 

After performing the ceremony, the minister created and signed a "minister's return," which was a slip of paper indicating that he had married the couple and on what date.  This testimonial was often written on the back of the marriage license itself.  It was then the minister's responsibility to drop the "return" off at the county clerk's office the next time he was in town, at which time the clerk would enter the marriage date into the county marriage book (the second of the two dates).  Sometimes, the minister failed to bring in the return slip, so while the lack of a second date may mean the couple never actually married (i.e., changed their minds at the last minute), it may mean they did marry but the minister simply didn't bring in his return. 

Here is an example of a Minister's Return (again, for Thomas Thompson and Delilah Davis) on the back of the marriage license. 

In some cases, each of the above steps might have been drawn out over a period of days (rarely weeks) or, in others, everything may have taken place on one day.  In the latter case, family and friends would show up at the county clerk's office, grant permission, post the bond, record the license, and have the county clerk, as JP, perform the ceremony — or they may even have brought their own minister with them.

There is often a third name associated with a marriage.  I am somewhat confused, myself, about this third name, which is usually entered in the book after the two dates.  In the case of a "Marriage Bond" book, the name of the third person is most often that of the person posting the bond.  In other county marriage books, the third name appears to be the name of the person who performed the marriage ceremony.  At other times, it appears to be the name of the parent or guardian giving permission or, even, the name of the county clerk making the entry in the book.  I think the point is that the third name may or may not be of genealogical interest, and you shouldn't jump to any conclusions about the relationship of this person to the couple or to the marriage.