At the 'Main' menu hold L1 and enter the following codes to unlock the corresponding effect. When you have entered each code release L1 and you will hear a confirmation noise if you have entered the code correcly.

Unlock Don's Big Head Goodie:
Enter Circle, Triangle, X, Square.

Unlock Challenge Map 2:
Enter X, X, Circle, X.

At the codes section enter 'DRDSS' without the quotes.

At the Password screen of the Options menu enter the following codes:

Alternate Costume (Raphael):
DMDML

Alternate Costume (Leonardo):
LDMSR

Alternate Costume (Michaelangelo):
RRLMD

Alternate Costume (Donatello):
DDSMS

Attack Power Up (Raphael):
RDSRL

Attack Power Up (Leonardo):
LMLSD

Attack Power Up (Donatello):
DRLDS

Defense Doubles (Leonardo):
MRLLM

Defense Doubles (Donatello):
SSSMR

Defense Doubles (Michaelangelo):
MRRML

Defense Doubles (Raphael):
LRMDS

Defense Power Up (Michaelangelo):
MSRMM

Change Sound Effects:
SMMRS

Power Up Effects Double (Raphael):
SLDSM/MSLLR

Power Up Effects Double (Donatello):
DMDRS

Power Up Effects Double (Michaelangelo):
MMSLR

Power Up Effects Double (Leonardo):
LSLSR

Unlimited Fire Shurikens/All Food is Pizza (Michaelangelo):
DRDSS

Unlimited Shurikens (Leonardo):
SMRDM

Unlock Playmates Toy Data Base:
LSDRM

Unlock Splinter for Story Mode:
MSRLS

Better attack for Leonardo:
LMLSD

Better attack for Raphael:
RDSRL

Better attack for Michaelangelo:
MSRMM

To unlock Shredder in Versus mode defeat Shredder as Leonardo,
Michaelangelo, Donatello or Raphael.

Unlock Splinter -
To unlock Splinter in Versus mode successfully complete Dojo Level 3 as Leonardo.

Unlock Evil Turtlebot -
To unlock the Evil Turtlebot in Versus mode successfully complete Level 3 as Leonardo, Michaelangelo, Donatello or Raphael.

Unlock Hun -
To unlock Hun in Versus mode successfully complete level 6 as Michaelangelo.