ParenthoodNJ.com

Bounce U - locations including Morganville, Piscataway, West Windsor
Gymboree - several NJ locations
My Gym - several NJ locations
The Little Gym - several locations in NJ

The Jersey Shore Children's Museum (Egg Harbor Township) - for children ages 1 to 10; hands-on activities and exhibits include a puppet corner, car factory, dinosaur table and a tot area 

Cutie Patooties (Englewood) - 3,500-square-foot play place for infants up to 6 years old; includes a 1,500-square-foot indoor play area; they also offer classes including yoga, art & music
New Jersey Children's Museum (Paramus) - toddlers and preschoolers love the interactive, hands-on exhibits, including a real fire truck, a real helicopter, cool fossil/dinosaur cave, play ballet studio, kids' medieval castle; there is a Baby Nook for infants & crawlers; they also host special events

Jelly bean Jungle (Marlton) - indoor play center includes ball pit room, carnival center with carousel, ocean center with pirate ship; designed for children up to 5 years old
Kid Junction (Mount Laurel) - recreational play place and indoor playground for toddlers and kids up to 9 years old. Features an indoor play area as well as a grocery store, pizzeria, costume shop and Kid Fit exercise room. You can also host parties here. There also is a separate play area for toddlers

The Garden State Discovery Museum (Cherry Hill) - a lot of hands-on activities, including a rock wall; they also host classes & programs as well as special events

Jumpnasium (Fairfield) - 13,000 square feet with more than 10 gigantic inflatables, basketball trampoline, air hockey and more; for ages 2 & up; some rides require min. height of 34 in

The Jersey Explorer Children's Museum (East Orange) - one of the first non-profit children’s museums in the country to display true-to-life, interactive exhibits that are the handiwork of at-risk youth; exhibits include Tomb of the Ancient Kings, Time Travel & more

Tiger Hall (Princeton) - 2,800-square-foot play and enrichment facility; offers classes, mommy and me sessions, kindergarten readiness programs, tutoring programs and workshops as well as daily open play opportunities

Jumping Jungle (East Brunswick) - a 10,000 square foot, indoor inflatable recreation center including 8 unique inflatables for open jumps and birthday parties, as well as kid-friendly arcade games. Children should be at least 34 inches tall but their are no age requirements
Uptown Playground (Kendall Park) - designed for children up to 10 years old and offers an indoor playground area and imagination stations. There also is Toddler Avenue, a separate area for younger children
jerZump (Monmouth Junction) - 17,500-square-foot facility geared towards children ages 1 to 7; includes a contained toddler room, an open play room with inflatables, arcade and snack bar. Open play on the weekends is geared for those over 34 inches and under 14 years of age. 
Let's Pretend (Old Bridge) - includes a bouncy house, ball pit, rock climbing wall, wii system and arcade as well as theme areas including Aloha Market and Rock Star Room; also offers "Date Night" for parents on Thursdays
Dynamic Fit Kids (Piscataway) - offers a weekly Klimbing Kidz program introducing children to fitness

Klimbing Kidz (Howell) - includes a 52-foot crawl-thru unit
JungleRRRific (Oakhurst, South Howell) - 4,200-square-foot indoor play gym for kids up to 8 years old; features include a rock wall, jungle-themed equipment and toddler area; two locations

Community Children's Museum (Dover) - an interactive children's museum for children 6 months to 10 years old.
This is a not-for-profit created by area parents. They also have a nice Kids Page.
Imagine That! Discovery Museum for Children (Florham Park) - 16,000 square foot hands-on museum featuring over 50 unique exhibits including a puppet theater, a pirate ship, a TV newsroom & more; they also host special events

Children's Museum of Somerset County - host special events for children at various locations in the county
Kid Junction (Branchburg) - recreational play place and indoor playground for toddlers and kids up to 9 years old. Features an indoor play area as well as a grocery store, pizzeria, costume shop and Kid Fit exercise room. You can also host parties here. There also is a separate play area for toddlers
Wee Play (Bridgewater) - a parent/child play program for walking children up to those 48 months old; classes are divided into 4 age groups
Boro Kid Zone (Hillsborough) - indoor play area includes a moon bounce castle, horizontal rock wall & 18-foot inflatable slide. Special programs include a drop-off play program; for infants up to 12 years old

Kidz Village (Kenilworth) - 11,000 square foot children’s play facility created for kids ages 10 & under; multi-level indoor playground & theme areas including comic book store, supermarket, dance academy, movie theater & 50's style diner; also offers special activities including storytime & pizza time