Sure you can. Check these links out:
Consider this as an example card:
https://trello.com/c/NIRpzVDM
You can link to a card as a photo using the following:
<img src="https://trello.com/c/NIRpzVDM.png" />
You could embed an iframe as follows:
<iframe src="https://trello.com/c/NIRpzVDM.html"></iframe>
You can also do a javascript that writes the card into your page:
<script src="https://trello.com/c/NIRpzVDM.js"></script>
which writes out some html using document.write("<html content>")
Another way is to request a card's JSON
https://trello.com/c/NIRpzVDM.json
All of these are available using a board as well...
https://trello.com/b/nC8QJJoZ.png
https://trello.com/b/nC8QJJoZ.html
https://trello.com/b/nC8QJJoZ.js
https://trello.com/b/nC8QJJoZ.json
In the lines of the solution by @bryan-larsen, I've created @welcomebot, a Trello bot that automates the process of getting people into a board.
In his own words:
I'm a bot designed to make Trello boards open for anyone to enter.
If you have a board where you want to let anyone enter, to comment and see and all that, just create a card, name it in any way you want (it can be something like "Welcome to this board, please comment here to enter!") and add me to that card.
Every Trello user who leaves a comment in that card will be automatically addded as a "normal" member to the board.
There's a test board where you can see the feature in action.
Best Answer
It's likely your browser's default setting not to print background. It saves on expensive printer ink.
You can change it though. If you select 'print backgrounds' in your browser's print dialog, it will include the background colors and icons.