There was a thread on RC Berwickshre site, Irregular marriages at Lamberton Toll, which explained it quite well. Runner posted it and I think it was very thorough.
Basically when Scotland became Protestant , the main church was the Church of SCotland any marriages etc outside COS was counted as "Irregular marriage"
Tom
First Secession, Second Secession, and Free Kirk marriages from the dates of these schisms were all "regular", as they were carried out according to the canon laws of these very presbyterian churches.
The meaning of "irregular" in this context is nothing to do with the church involved as such, but everything to do with Scots Law, there being 3 forms of irregular marriage in Scotland which survived to 1939, with 1 of the 3 actually surviving to 2006 (or was it 2005?). One of the FAQs on the TalkingScot DG goes into the subject in detail.
A couple who were communicants of the Established Church of Scotland could still and frequently did, undergo an irregular marriage; although the kirk session would give them h**l when they found out, which they almost always did.

ibi