A space. First of all, obviously, you need a space to workout! But - I think it's important to note that it doesn't have to be a dedicated space solely for your home gym. I do most of my home workouts in the basement - this is what that looks like:
Super neat and motivational exercise environment, huh?! I suppose if I put some effort into organizing (and keeping organized, which is really the key) the basement, I could have a portion that was just for exercising and keep equipment laid out ready for use (and maybe add some aesthetics beyond a concrete floor). But as it is, it's really not that hard to pull out what I need, and yet keep it out of the way of other uses of the basement the rest of the time.
My bike (when not using it frequently outdoors, which I haven't since before being pregnant) stays hooked into my bike trainer, which allows me to easily flip it up against the wall. (When it's pulled down, it blocks access to both the litter boxes to the left and some of the storage to the right.) And easily flip it down and put the front wheel on the stand to ride it indoors.
The weights and rolled up mat just hang out to the side of the washer/dryer, the mat generally on top of whatever boxes are currently stacked in that area. While working out, the mat goes in the main cleared out space of the basement, which is directly in front of the washer and dryer.
I do at least have some race memorabilia hanging in the stairwell to make the space somewhat more workout-oriented and maybe even give me a boost of motivation when needed.
One exception though is that I usually do yoga (on a nicer mat that's kept just for yoga - I'm not going to take the one that sits on the basement floor where kitty litter gets tracked all over the place into a yoga studio) in the living room (or sometimes in the baby's room). For yoga I think there's more benefit from having a nicer environment around you! The rest of the house is definitely more zen than the basement.
My computer. There's so much available online you can do for free or cheap, and without leaving your house (the latter being particularly important now with a baby!). I've paid, at times, for subscriptions to DailyBurn and YogaDownload.com; I've also found a lot of stuff for free on Youtube - some Jillian Michaels videos and a yoga channel I'm really liking currently. I also watch netflix when riding my bike on the trainer (my laptop nestles perfectly on top of the handlebars and brake wires).
Adjustable hand weights. With limited space, not to mention having to store them out of the way in between workouts, I think this style of dumbbell is perfect. You don't have to pull out every set if you're using different weights within the same workout - you just have to store the one pair, and adjust as needed for each exercise. Plus, the total set, which really gives you 5 pairs, is not much more than buying one individual pair that you'd have to replace to use a higher weight.
This set is 5-lb increments from 5 to 25 lbs; I also used to have one that was 2.5-lb increments, from 2.5 to 12.5 lbs (that one is now in my mom's possession for the 30 Day Shred workouts - she's moved up from the 2.5 to the 5 pounders!); I found both at Target.
Doorway pull-up bar. Probably the gym equipment that seems hardest to replicate at home is the assisted pull-up machine. I definitely can't do a real pull-up! (Not anymore... I was getting to be able to just barely eek out one last year after doing NROLFW.) So just a bar isn't enough. But a bar, which is cheap and fits securely in a doorway without any installation (out of the way - unless you're 6'4" but Abe just ducks his head), plus resistance bands to assist (you can also put a chair underneath for assistance or do reverse pull-ups) give you a lot of options to build upper body strength.
Rec center membership. Yes, I do most of my workouts these days at home - but sometimes you need another option. For swimming and the occasional treadmill run or alternate cardio (stair stepper, most often), I buy a monthly pass to the local parks & rec district facilities.
The drop-in pass is only $30 a month (or cheaper if you have more than one family member or buy for more than one month at a time), and gets you access to all of the aquatic and cardio/weight room facilities - which is several locations. There are some by home and some closer to work. Cheaper than all but the cheapest gym, especially for multi-location access.