At 13 month old, my daughter love:

Board books with baby faces
Shape sorter cube thing
Stacking ring
General household items (random bookmark, empty milk bottle, mail, you get the idea)

Pounding bench. Melissa and Doug makes a good one that is four wooden balls, about the size of ping pong balls, that then click clack down through a ladder. Both our kids loved it. We put it on a tray to keep balls from rolling under furniture.

Megablocks
Small bead maze activity cube
Imaginarium at toysrus and B. at target have really nice toys
Books about trains or whatever he's interested in

Baby Einstein take along tunes. Mine loves it. Has learned the word "music" so I say music and she points to it and tries to press the button.
Block and large Tupperware so Shea n take them in and out
One of those beads on wire puzzles. Melissa and Doug have one with suction cups so it'll stick to the floor or high chair.
Tons of board books. Get ones that have little animals and cut outs with textures like fur
Favorites: if i was a rabbit, brown bear brown bear what do you see.

My 14-month-old likes stacking toys like the one suggested in the link above. He also likes the shape-sorting toys, and basically anything his older brother has Surprisingly, he seems really into books about animals and loves to have me read him books where I point at animals and make animal sounds.

My 13 month old likes:

Hape cube
Board books (especially 100 first words and the ones in that series)
V-tech sit to stand walker
Anything that requires putting something into something else
Large wooden puzzles with little handles
Ball pit (kiddie pool filled with little plastic balls)

I have 13 month old twins. Their favorite toys are little people, putting spoons in a washed orange juice jug, and this thing: http://www.target.com/p/fat-brain-toys-spinagain/-/A-51960594?ref=tgt_adv_XS000000&AFID=google_pla_df&CPNG=PLA_Toys+Shopping&adgroup=SC_Toys_Top%20Performers&LID=700000001170770pgs&network=g&device=c&location=9061283&gclid=CjwKEAjwkq7GBRDun9iu2JjyhmsSJADHCD_HZ48ZNlOs29FmHNZtILmKR0u2gOH54y0tK1UW4J_6TBoCWC7w_wcB&gclsrc=aw.ds

They also like anything with buttons that plays music.

He's a bit behind in development since he was premature. Any ideas? Thx.