Hey zaba, long time no see!
The first thing to keep in mind is that Saussure and Peirce were describing completely different things. Saussure was describing an ontology for the sign, which consists of two co-extensive parts: the signifier and the signified. Peirce, on the other hand, was describing a process of semiosis. In order to understand Peirce's sign, you have to think about it as happening over time.
So, the three parts:
the sign (or, more often, representamen): the thing that represents the denoted object; similar in many respects to Saussure's "signifier".
the object: the mental conceptualization denoted by the sign; includes everything in Saussure's "signified", but also more.
the interpretant: the interpreted consequence of the sign-object relationship, which itself can become a sign (this is the big one).
That's the typology, but here's a very simple example: I'm standing by the fridge, eating a sandwich I found. You walk in and yell, "THIEF!".
Phase one:
Representamen: the phonetic sequence [θif]
Object: the word "thief"
Interpretant: your intention to reference a thief
Phase two:
Representamen: your intention to reference a thief
Object: an existing thief, possibly me
Interpretant: you are calling me a thief
Phase three:
Representamen: you are calling me a thief
Object: you are angry that i have stolen your sandwich
Interpretant: your intention to confront me over issues of sandwich theft
Phase four:
Representamen: your intention to confront me over issues of sandwich theft
...and so on.
Does that clarify at all?