Are there any bridge systems where cards played convey deeper information in a manner similar to bidding systems?
For a starter, it is obvious that such signalling only applies on defence - Declarer has no need to communicate with his partner (Dummy), and such communication is anyways illegal. Let's start with the basic defensive systems and progress from there.
 All of these are perfectly legal when fully explained to the opponents (except some of the more complex techniques are restricted in limited-master-point events).


Basic Signals:

Lead highest of touching honours (except King from Ace-King) either at the top of a suit or from an interior broken sequence; fourth-best from length; highest of a doubleton; low to show interest in having the suit lead back and high to deny interest in having the suit lead back; Lead high for partner to rough asking for a higher-ranked side-suit lead back an low to ask for a lower-ranked side-suit lead back.
Follow in trumps high-low on first two rounds to show a 3rd trump and interest in a rough; follow high-low to partner's lead suit to express interest in continuation, low-high to ask for a switch; follow high-low in other suits lead by Declarer to show an even number, low-high to show an odd number; Follow high-low as suit preference for a higher-ranked side-suit and low-high as suit preference for a lower-ranked side-suit.



You will of course notice that many of these are contradictory or ambiguous - that is the cost of asking a very restricted language to convey a rich amount of information. Established partnerships discuss the priority of these signals at great length, but the general gist is (usually): Attitude first; Count once attitude is known; Suit-preference once count is known, or when obligatory. Deciphering that consistently with your partner is one of the most highly rewarded skills in Bridge, as most of your opponents are so very bad at it.
;-)

More complex signals involve adding a mixture of these variations to the basic scheme:


Second-highest from touching honours. Sometimes only at the top; sometimes only from interior sequence (Coded 9's &amp; 10's); sometimes both.
Upside-Down signals - reversing the meaning of high-low and low-high sequences;
Odd-Even signaling - replacing high-low and low-high combinations with the play of either an odd-spot or an even-spot, sometimes combining attitude with suit-preference so that an odd-spot encourages; a low-even spot asks for a lower side-suit switch; and a high-even spot asks for a higher side-suit switch. 
Round-the-corner suit preference in place of absolute suit-preference.


That's all I can recall off the top of my head, but I have seen attempts to play even more unusual combinations. Myself, I don't think the benefit is worth the risk of a disaster from misunderstanding.

Update:
With some exceptions, any partnership agreement on the meaning of a legal call made or card played, or sequence of such, that is well explained in advance of the round/match to the opponents is legal. The exceptions generally cover methods deemed too complex, in the context of the competition being held, for the opponents to defend and understand. Generally there are few restrictions in 10-day World Team Championships, and very many in 3 hours novice match-point games.

The section on Carding on each chart of current ACBL Convention Charts explains the defensive carding systems allowed or disallowed for the corresponding class of competitions held under the aegis of the American Contract Bridge League.

Update #2 - Partnership Agreements:  


re Duplicate Bridge: see Law 40 in Section 6, pages 48-52 of Laws of Duplicate Bridge
re Contract Bridge: see Laws 16 (page 10) and 40 (page 20); Proprieties Points 2 (page 45) and 4 (page 47) of Laws of Contract Bridge


Update #3: A sample Convention Card (pdf) as approved for ACBL sanctioned events.