In English
heraldry the words
mantling and
lambrequin mean the same thing, but they probably have two different origins, and are still distinct in Continental heraldry. It is usually called mantling.
A full coat of arms has a shield in the centre, and most have a helmet perched on top of the shield, and a crest perched on the helmet. It may also have a ground to stand on and creatures to support it at each side. Behind the helmet is a swirling, ragged display of what look like large ribbons disporting themselves in whatever arrangement the heraldic artist thought looked good. This mass of ragged cloth is the mantling or lambrequin.
The lambrequin was a cloth hanging down to cover something. Specifically, it was the cloth that Crusaders wore over their helmet to keep the heat down a bit. Think of modern Arab headgear and the French Foreign Legion cap.
It was found that someone in the thick of battle, taking a lot of blows on the head, was somewhat protected by the entangling effect of the cloth. So a lambrequin cut to ribbons could be seen as a battle honour. This was represented in the heraldic art of the Middle Ages.
In later centuries the display of a coat of arms got ever fancier. The plush and usually ermine-lined cloak (mantle) of the rich and noble (called the robe of estate) made a good background for their arms. In Continental heraldry many royal coats of arms are displayed fully on an entire cloak, hanging from a crown, pinned back like stage curtains. In German this mantling is called the Helmmantel, while the ribbony lambrequin is called the Helmdecke.
In English heraldry the mantle or robe of estate is never used. The mantling is always depicted as the slashed lambrequin, though the space it occupies is much larger than the original small cloth it must once have been. However, the colours of the mantling tend to come from the robe of estate, because ermine is very common, and the Sovereign uses cloth of gold.
The mantling hangs from the top of the helmet, underneath the crest. (A crest is not a coat of arms, it is the Crusader equivalent of a nodding dog or furry dice on top of it!) It is attached to these by the wreath or torse, a small twisted loop of cloth. Think of the rope-like circlets that hold Arab headgear in place.
In cases where the crest was an animal, the mantling was sometimes depicted (in early heraldry) as a continuation of it: e.g. a lion's flayed skin if the crest was a lion's helmet, or a monk's robe if the crest was a monk.
Fashions for what colours to use have changed and fluctuated considerably over the centuries. There is a general tendency to use the principal colours of the arms: the main tincture (red, blue, green, etc) on the outside of the mantling and the main metal (silver = white, gold = yellow) for the lining. But red mantling lined with argent (silver) became very common, regardless of the colours of the arms. Lords tended to use ermine linings (ermine is argent with small black tail-markings). So gules (red) mantling lined with ermine is common. From the time of Elizabeth I the Sovereign and the Prince of Wales have used gold mantling lined with ermine, and no-one else is permitted to.
The expression used in England is 'lined with'. In Scotland they say 'doubled with'.
The mantling occasionally has some extra decoration on it. This usually is just decoration, with no heraldic meaning: fine patterning scattered over it, or floral backgrounding, a bit of artistic licence; but occasionally a simple geometric pattern such as stripes is blazoned (gazetted or officially specified). In a few rare cases the two sides (left and right of the helmet) are different colours.