Probably the most famous symbol in Agamemnon is that of the "net." This image appears at numerous points in the text, most memorably when Clytemnestra appears outside the palace at the end of the play, standing over the dead bodies of Agamemnon and Cassandra; there, she boasts about how (in Collard's translation), "A net with no way through, just as for fish, I stake out round him, an evil wealth of clothing" (1382-1383). But similar images appear throughout the play, such as when the Chorus says to the dead Agamemnon, "You lie in this spider's web breathing out your life in a death which is impious" (1492-1493), or even in the famous image of the purple fabrics that Clytemnestra bullies Agamemnon into trampling on as he walks into the palace. Taken together, these images of nets, spider webs, and entangling clothing create a common image of Agamemnon's death inexorably closing in on him. Could this also be an image of the inescapable power of fate? That would depend on how you interpret the play's treatment of the theme of "Fate and Free Will," and we're not going to spoil your fun.
Family is a theme in Aeschylus's play, but it is also one of recurrent symbols or motifs. We touch on this a little bit in our section on the theme of "Family," so you can look there for more details on the specific passages where this motif appears. Basically, though, Aeschylus keeps the theme of family fresh in our minds by describing many things that we would not typically think of in terms of family relations in language derived from genealogy. Thus, on several occasions, characters refer to the night that has just passed as having "given birth to" the present day. In a play in which the crimes of previous generations of Agamemnon's family are rehashed and repeated in the present generation, it makes sense to think of the simple passage of time in generational terms.