I'm an exercise equipment junkie. I always have a list of things to buy - both instructional books and DVD's and actual equipment - and a bedroom filled with things I've picked up over the years (most of which I actually use regularly). I recently posted a comment about wanting some new pieces and a friend of mine replied that the only equipment one needs to train is one's own body. That got me thinking.
First of all, you do not need any equipment at all to train in karate. There is no aspect of fitness that cannot be trained using your own body, gravity, a little space, and an appropriate training method. Since you do not need any equipment, you certainly do not need any particular piece of equipment. Anybody who says you "can't get in shape without product X" is probably trying to sell you product X, or has drank the Kool-Aid offered by people who do sell it.
Having said that, training with your own body and gravity alone is not necessarily the best way to develop all aspects of personal fitness. A great example is hip snap - a powerful movement through the last few degrees of hip extension, using the glutes primarily, that drives punching power, closing speed, and jumping. Can you develop good hip snap with your own bodyweight? Of course. However, a program of heavy kettlebell swings will develop your hip snap to a greater degree in less time, and it might develop that quality to a greater degree than is even possible with bodyweight alone. I can say similar things about many other implements. Can you become a powerful striker without ever hitting anything? Maybe (maybe not - that's another argument). But using a makiwara or a heavy bag will make the task much, much easier.
If you train at a school that uses equipment, that's great, especially if there's a small group or a lot of equipment! I suspect that most dojos have only a limited amount of equipment, and you may not get as much use out of the items you do need in a group setting as you would prefer. You should also be training more often than just at school - strength and skill training should both be done with frequent, brief sessions, and I doubt you're going to the dojo three times a day for five minutes at a time unless you're a full time martial arts instructor. The solution? Buy some stuff for home.
What equipment should you own? As with all things, that depends on several factors. First, identify your training needs. If your biggest issue is flexibility you might want different stuff than if you need a lot more strength. If you're stiff and sore all the time your first purchase should be a foam roller.
Second, identify your limiting factors. Money is a big one for most of us. If money is very tight, consider some do it yourself ideas, like the ones you can find on Ross Training. You can make very serviceable versions of a lot of pieces of equipment with a little time and a little knowhow. If space is very tight, consider a gym membership - some gyms have much better lineups of equipment than others. Some items also take up next to no space - think bands, as opposed to barbell and plates. Check your flooring - if you work out in the upper floor of a regular house, like I do, you might not be able to do some exercises, like anything where you throw or slam a medicine ball. That's why I don't own any medicine balls!
Once you have a rough budget and an idea of what kind of room you're working with you can start shopping. I highly recommend making a list and pondering it for at least a week, if not longer, to avoid impulse buying (maybe that's just me).
You could group equipment into various categories.
Resistance is equipment that resists movement or loads positions. Think weights (dumbells and barbells), bands, kettlebells, clubbells, medicine balls, and I'd even throw chinup bars or suspension trainers in there (because they let you use your body to load different movements). Resistance equipment is generally going to help you get stronger.
Striking surfaces are things you can hit. Makiwaras, striking bags, heavy bags. Some are more for teaching accuracy (speed bag), some for power (heavy bag or re-breakable boards), and some for body conditioning (makiwara). Every martial artist needs to spend some training time hitting things instead of striking air - the body mechanics are very different, and you can't really learn to find your range without hitting something at least some of the time.
Unstable surfaces are things you can support yourself on to train balance and increase core activation. Bosu Balls, exercise balls, airex pads, and slack lines are all good for this. Probably shouldn't be the first thing you buy.
Recovery aids are undervalued in the training community. And no, I don't mean anabolic steroids (that's a subject for another post), I mean items to help you improve recovery at home. Basically this covers a variety of self-myofascial release devices, or self-massagers. Foam rollers, sticks, tennis balls taped together - they can take a variety of forms. I can't tell you how much foam rolling my legs helps my recovery when I'm training really hard.
Flexibility aids are those items that will help you stretch out. You've probably seen the ads for the leg spreader - that metal contraption that winches your legs apart to push you towards a full split. Seems like a poor investment to me, in both space and money, but I've never used them.
Cardio equipment is stuff that helps you work for endurance. Most resistance equipment can double as cardio - you can lift a dumbell for strength, but you can also power snatch a dumbell for sets to build up your cardio. I'd put a timer and a heart rate monitor into this category, along with rowing machines, treadmills, stairclimbers, etc. A timer, something like the Gymboss, is a great idea, and I suspect there are good uses to be made of a heart rate monitor, though I haven't gotten into that yet. Jump ropes are also great and underappreciated.
If you don't have any specific priorities I have some suggestions. (I'll assume only that you live somewhere with gravity and at least a small space for training.)
I think most of us would benefit first from some way to train for pulling strength. Karatekas tend to do a lot of pushups, which train the front part of the shoulder girdle. You need pulling strength to balance the forces around that joint and prevent injuries. A chinup bar or a Door Gym type of device is cheap and a good start.
Next you should get something to hit. Here space is a much bigger issue. For hand conditioning you can get a portable makiwara (I love mine - bought it at http://www.fightingarts.com/, which I should mention for disclosure reasons is owned by my former sensei, although he doesn't know I shop there). Cheap and doesn't take up much space. After that a heavy bag is a nice purchase, or maybe something like the Body Action System (I've never used one, but I find them intriguing, and it's on my short list for a Christmas present).
A Gymboss timer and a foam roller are great $20 investments. I love using the Gymboss to organize all my interval training - I'll use a watch in a pinch, but the Gymboss is much better for hard sets (the alarm can be set to sound at any interval or combination of intervals, and hearing the "beep" is easier than checking a watch every 2 seconds when you're struggling to breathe). A foam roller will aid your flexibility but also really helps with recovery if you're working close to your own limits.
I like kettlebells and selectorized dumbells quite a bit and use both in my training every week. I'd advise getting one kettlebell in each weight category - I got a pair of 35 lbs. and a pair of 54 lbs. and I almost always use just one of them. I'd much rather have a 35, a 53, and a 70 (and maybe an 88) instead of the ones I got. They're pricey, but I really like doing presses, getups, and swings with them, and you can easily build your entire strength and conditioning program around just a couple of kettlebells and a park.
There are more advantages to having equipment at home. One is motivational - for many people having a shiny new gadget in the house makes them want to use it. Can't let dust accumulate on the new makiwara! Another is convenience. If you can't get to the dojo because of work or family constraints or an injury you can probably rattle off a few pullups just before hopping into the shower.
Focus on tools that are multi-use, can be used without help (a disadvantage to focus mitts), and if you're at all handy, think about some DIY projects. You m ay not need anything to make progress, but many of the things you can buy can help your progress.