Q:

Why is the European robin associated with winter?

Reading some of the comments on this answer:

....they're seen in the UK as winter birds. Not, I think, that they're
  not around in the summer; more that other birds aren't around in the
  winter but robins are. Robins are often found on Christmas cards in
  the UK. (Christmas being in the winter in the UK.)

Robins are around all year in the summer, spring, etc. and have a red breast, etc (except juveniles) all year. RSBP web site confirms this:

The UK's favourite bird - with its bright red breast it is familiar
  throughout the year and especially at Christmas! Males and females
  look identical, and young birds have no red breast and are spotted
  with golden brown. Robins sing nearly all year round and despite their
  cute appearance, they are aggressively territorial and are quick to
  drive away intruders. They will sing at night next to street lights.

So why are robins so associated with Christmas and winter? Where does this come from?

A:

Robins are an all year round bird in the UK:
RSPB

with its bright red breast it is familar throughout the year and especially at Christmas! Males and females look identical, and young birds have no red breast and are spotted with golden brown.

So this proves, for those of a keen eye, that there's inherently nothing special about the Robin Male vs Female, only the juveniles have a different colouring, no changes throughout the seasons like some mammals, nothing. They're pretty dapper in their red chests all year.
SoWhyDoI

Although robin folklore existed for many years, the robin became far
  more widely associated with Christmas after it was depicted on
  Christmas cards in 19th century England; a tradition that has been
  retained to this day.
Why are robins on Christmas cards? It’s all inspired by the robins who
  used to deliver the Christmas cards in 19th century Britain.
In the 1800s, British postmen wore bright red uniforms. They wore red in honor of the crown since red is considered both a Royal color and an important color on the English flag. (Incidentally, this may also be one of the reasons why British post boxes were eventually standardized to be red). The postmen in their red-breasted coats resembled the much-loved British bird, the robin red-breast, earning Victorian postmen the nick-name: Robins.

Ok so this is a nice piece of information I did not know - Posties have always had a nickname of sorts, and it could be a valid reason for why they are on Christmas cards etc around that time of year, however there are other speculations too.
Christianity:

Legend has it that the robin's redbreast gives it a direct link to Christianity.  
One fable suggests that when the baby Jesus was in his manger in the stable, the fire which had been lit to keep him warm started to blaze up very strongly. A brown robin, noticing that Mary had been distracted by the inn-keeper’s wife, placed himself between the fire and the face of baby Jesus. The robin fluffed out its feathers to protect the baby, but in so-doing its breast was scorched by the fire. This redness was then passed onto future generations of robins.

Further reading: Saga website on Christianity and Robins
I think as well one good idea about why they are on postcards is when we used to get a huge blanket of snow Mr Robin Redbreast would stand out against the bland colours of winter with his bright red feathers, the contrast looked good and someone painted it and... here we are!

A:

One unusual thing about robins is that they're territorial all year round. Combined with the way they sing from prominent spots to mark out their territory, this makes them really rather obvious in the winter, when other birds are quiet and inconspicuous.
Robins, as well as being plentiful, are also quite confiding around humans, mainly in the UK where we lack large wild herbivores to turn over soil and expose insects1.
While mainly insectivorous they'll happily take seed and other food from human sources, meaning that when food supplies are low they come to gardens.  Once there, as they're so confiding, they often feed closer to people than other birds, coming for food almost immediately.  It's not unknown for people to hand-feed wild robins in the garden.
Overall if you hear a bird singing around Christmas it's probably a Robin, and the song will draw attention to a bird which is after all quite striking for something so common. Their habit of perching on artificial structures for some time (and returning to the same song posts) also makes them easy to observe, and paint/draw/photograph. 

1 Wikipedia cites RSPB Birds of Britain and Europe, Rob Hume 2002 (ISBN 0751312347), which is where I'd read about this peculiarly localised aspect of behaviour.

