I think the term was first coined by a guy called Wilfrid Bion, who was an early 20th century British Psychoanalyst.
He suggested that when we are very young, we rely on a close parental figure, usually the mother, to help us regulate our own emotions - we cannot do it ourselves and rely on the parent to help us deal with experiences that would otherwise feel overwhelming. He called this containment.
The idea is that the parent can hold on to these overwhelming feelings and then 'give them back' in a more bearable form that isn't toxic or as threatening. Containment doesn't have to be verbal - it could be being soothed by being held, or wrapped in a blanket. Or it could be talked to soothingly, even before we understand words.
If the parent is, for whatever reason (depression, anxiety, money worries, or not having had the experience of being contained themselves) not able to help contain overwhelming experiences, the infant or child is not contained and their view of their parent, themselves and the world are shaped as a result.
This is where therapy can come into play as it can offer people the opportunity for people to feel the sort of containment they may have not experienced as very young children. It cannot replace what we may not have had but it can help us come to a point as adults where we can care for ourselves.
Sorry, bit theoretical but that's where it comes from. Basically what Catalyst said ;-)
If you are feeling exceptionally nerdy there is a
useful presentation on the concept - slide 8 onwards deals with containment.