Get your mind out of the gutter. I’m not only talking about sex and romance. Those are a given, well they are supposed to be.
I’m talking about intellectual stimulation and physical activity.
Is it important that you can relate to your partner on a intellectual level, being able to discuss politics, art, literature, music, etc. into the wee hours of the morning.
Or, having a great activity partner who you can hike, travel, play sports, camp, ski, boat, etc. with?
Both?
Neither?
I think both are important (and I voted for both), but not necessarily because I need someone who constantly wants to be running around the great outdoors. I think it’s important that two people in a relationship aren’t terribly dissimilar in their levels of activity. That’s the sort of situation where one person is going to get extremely bored or feeling cooped up, or the other will feel like they are constantly being dragged everywhere. Nobody wins in that scenario.
And I think it’s a given that two people should stimulate each other intellectually – again, by whatever definition fits that for them. Whether that means discussions about the Japanese rationale for Pearl Harbor, or a poetry collection, or the 1989 World Series, or the last season of “24” – so long as both people are feeling stimulated in similar ways. But it’s absolutely important to me, and I think it’s a vital component of any relationship.
What Asimovian said. Both are important to me, but not equally so.
By physically stimulated, I feel no need to date someone who absolutely must go rock climbing every weekend, but I need someone who likes the beach as much as I do. Intellectually I don’t need someone I can discuss string theory with (would be nice though), but I need someone with whom we can energize each other.
I think that there is a third level of connection, emotional, and even a fourth, spiritual. An atheist would be best for me, but I’d rather date someone slightly religious than a complete cynic.
The other week I went out with a priest. Our conversation stimulated me on the spiritual, intellectual, and emotional levels. I can’t describe how awesome it was. We decided to just be friends, but man, she is a very cool friend!
Speaking as someone who is with a partner that does not fulfill either of these categories - Oy!
Has to be both.
From an activity standpoint, I think a partner needs to be similarly interested in activity as you are. Not necessarily the exact same ones, but I think a similar willingness to try new activities/sports is important. I also think being willing to slob out at home once and a while if your partner is feeling lazy is important - I’ve dated guys that constantly need to be doing something and it’s fun, but can get very tiring. Sometimes it’s nice to bum around the house in your PJs all day watching stupid re-run movies on TV.
Intellectual - well, that’s kind of a given. If you can’t talk to your partner about stuff that’s important to you, why on earth are they your partner? A nice ass? A big shwang? Lord would that get old fast.
I think its important the person is open to what I am interested in, and I feel open to what they are interested in.
You can go out and try and find someone who is some doppleganger of yourself. I used to want this, because my best friend and his wife are basically gender-opposite clones of each other and share a ton of hobbies and I wanted a partner like that.
But you know, I dated many women that shared my hobbies…and I was actually kind of put off. I guess I learned that for me, I want someone who has something NEW or UNIQUE that I can explore. Think of a friend who got you into a cool hobby you never would have known about yourself- that’s what I really enjoy. I guess its kind of off-topic but ultimately what I am saying is that its important for my partner to stimulate my interest in learning new things, and be open, in turn, to me introducing them to my existing interests. 
Well it depends if you also provide intellectually stimulating points and are physically active. While it is important that people have lives away from their partners, you have to have things that draw you together…outside of the bedroom. Level of intellect and physical activity are important factors of our lives, and for two people to be successful partners over a long term, I would believe you need that compatibility.