[beaudeal]

For me, the most important trait I look for in a co-founder is trust. If that element is absent, I think building a long-term partnership would be too difficult, no matter how wonderful they may be in other areas. They also need to be reliable, as you need someone who's going to stick it out through the highs and lows. Apart from that, I look for an overall high level of intelligence and someone who compliments my skill set (e.g. someone with a killer sense of design).

[johnm]

As you say, trust (or, more specifically trustworthiness). This ends up being a lot in terms of integrity and authenticity.

Conviction -- particularly courage of their convictions.

Understands the difference between fact and opinion. Also known as the Criteria of Reality.

Argues honestly. Especially when combined with conviction and criteria of reality, this allows for the flexibility to to learn and adapt rapidly.

[michael_dorfman]

Trust, but verify.

No matter how much you trust your co-founder (and you should definitely choose someone you trust a great deal) make sure that there are adequate legal documents in place to protect your interests. If, years down the road, the person you trust ends up not being so trustworthy, you don't want to end up screwed. (I speak from painful experience....)