I have seen some 65 year olds that exercise with just as much vigor as 30 year olds. It all depends on the lifestyle and level of activity that the person is used to.
While I do recommend low impact exercises generally, not just for 65 year olds, but for most people, that doesn't preclude 65+ year olds from being able to do more intense and high impact exercises like running, cross fit, lifting weights, etc. I do recommend low impact because it does keep the pressure off the joints and still improves the joint mobility, and maintains some bone strength as well.
The first low impact exercise that comes to mind is T'ai Chi. It's a multi-functional exercise. It builds flexibility, and strength, but it also centers and focuses the mind, which is also great for the 65+ population. T'ai Chi is what we call a yen form of martial art where there are ultra slow movements and positions (called forms) that are done over a period of 60 to 75 minutes. By using just the body as a resistance tool, you increase balance, strength, flexibility, and that centered peace of mind feeling. I would recommend either a T'ai Chi or Qigong, or a yen form of martial arts.
The second type of low impact exercise is a yoga. I recommend a restorative, flow type yoga. I practice a style of yoga called Ashtanga, and you'd be surprised how many 65+ women and men visit that classroom, and are quite advanced practitioners. If you are a first time practitioner, I do recommend going to a beginner class. Most teachers know how to show you the modifications of specific yoga asanas, or postures, so that it fits your level of practice, flexibility, and strength.
The third thing is swimming. Swimming is a great no impact, strength building, AND cardiovascular exercise. If you have a pool, you can just jump in and do a couple of laps. If you have a local gym or YMCA that holds water aerobics class, those can often be fun as well. I definitely recommend water aerobics focus not only on different aspects of non-weight bearing strength building exercises, they incorporate cardiovascular component in a group setting. Classes are always more fun!
The other thing you can do is cycling. Just because you're 65+ doesn't mean you can't get on a bicycle and ride your way down a trail, or even if it's a stationary bike - it builds strength and endurance and again, it's no impact and it's great for people who suffer from arthritis or back pain of any kind.
Similar to swimming, if you have arthritis or tend to be a little bit overweight, and that hurts the joints and you can't do other low impact things like walking, or any kind of machine work such as elliptical, then water aerobics is actually the perfect option for you, because there is no impact on the joints at all.
I also recommend as a low impact exercise, dance classes. There are a plethora of types of dance classes you can choose from. A beginner's Zumba class, which tends to be pretty low impact, moderate intensity. I would recommend a Zumba class for someone who is somewhat in shape, because some Zumba classes can go a little fast. If you are a dancer of any type, then Zumba might be a choice for you. There may be other dance studios in your area that offer things such as ballroom dancing, waltz, or tango. These are activities that still build that cardiovascular endurance, to some people's surprise, but are also fun, and sometimes require a partner, so there is some relatedness, or intimacy there. It can be used as a great date for a husband/wife couple, and can keep you in shape at the same time, and don't require too much flexibility or impact on the joints.
Those are some great low impact exercises for folks over 65, and I also recommend that if you are serious about improving your strength, flexibility, and endurance, that you look into a personal trainer that specializes in training those of an advanced age. There are other types of exercises that I'm not aware of, that's outside of my view and expertise, that I'm sure personal trainers have that are fun and effective.