I received this reply last month from the UK. I had written to the Cambridges back in April for Louis & Princess Charlotte's Birthday and the Cambridges' 10th Anniversary.
The front of the envelope is the usual from the British Royals, featuring the postmark from the Court Post office at Buckingham Palace.
The back of the envelope is the usual from the Cambridges, featuring Kensington Palace under the coronet of the child on the heir apparent.
Inside are three photo cards, one for each event.
The first card is for Prince Louis's birthday. It features a photo of him taken by Duchess Kate on his first day of nursery school. This photo was released to mark his birthday.
The Duke and Duchess of Cambridge are most grateful for the very kind message you sent to Prince Louis occasion of his third birthday.
This was greatly appreciated by Their Royal Highnesses who send you their best wishes.
It looks like there was a little mistake in the card. The first sentence doesn't quite make sense grammatically, it should be something like "[...] Prince Louis on the occasion of his third birthday. "
The 2nd card is for Princess Charlotte's Birthday. It also features a photo by Duchess Kate that was released on Charlotte's birthday.
The Duke and Duchess of Cambridge appreciated your kind thought in writing to Princess Charlotte as you did on the occasion of her sixth birthday.
It really was most thoughtful of you and greatly appreciated by Their Royal Highnesses who send you their best wishes.
The 3rd card is for the couple's 10th wedding anniversary. And features a photo taken by Chris Floyd at Kensington Palace to mark the occasion.
The card says:
The Duke and Duchess of Cambridge were touched by your kind message on the occasion of their 10th wedding anniversary.
This was most thoughtful of you and greatly appreciated by Their Royal Highnesses who send you their warmest thanks and best wishes.