What is the most effective way to kill weeds that does not involve digging or killing other plants?
I assume due to your comment that this a garden you are worried about.
You can put down a heavy application of mulch around your rows of crops, you still will have to manually weed, but the mulch will prevent some weeds from coming up and will make those that do easier to pull. Even better, this in conjunction with ground cloth works wonders because the ground cloth prevents seeds from finding soil and those underneath from getting through to the sun. Neither of these solutions requires digging, just hard work. Because you are only doing the no digging method here you will get weeds in between your crop plants where the ground cloth didn't cover, but the mulch in these areas will keep it down more than no mulch.
Now if you want to get serious, and never weed again in your life, you cut out pieces of ground cloth that fit your garden and cut an X in the cloth where a plant stem will be, this way the only space not covered by ground cloth is where your crops are growing. Then cover with 2-3 inches of mulch.
I use this exact method for my rose gardens and my herb gardens.
The best method I know of doing weed control without weeding is sheet mulching, but this is best if applied before the weed is outgrowing the vegetables, making it a very labor intensive technique.
If this was a lawn, I would suggest a weed spray. There are weed sprays from Ortho and Spectracide that will kill weeds without harming grass. Note that this is different than something like Roundup or GroundClear, which simply kills all vegetation.
For a garden or flower bed, weed sprays will often harm your plants too (after all, they're not grass). However, you can check the label of the sprays to see what does/does not kill. You might get lucky.
If the spray won't work, it sounds like you need to do some manual labor. Start digging/plucking.
As a future preventative, you can apply a weed preventer like Preen to your garden. Preen stops new weeds from germinating, but it does not harm established plants, so it is good to use in an already-weeded garden.
Others have already suggested landscape fabric and mulch, so I won't repeat that one. You can use Preen in combination with this, just sprinkle some on top once the garden plants are established.
The most effective way to kill weeds is to prevent them from popping up in the first place. You mentioned in a comment that you are weeding around cabbages, so this answer is geared toward weed control in an annual vegetable garden. (I don't use herbicides, so I don't mention that below.)
See my answer to a question on controlling ragweed for some prevention techniques.
It's also worth noting that some control techniques will work better on certain weeds than on others. Weeds vary in their behavior: some are creepers, some set tons of seed quickly, some will send out runners even under a sheet mulch. The advice that follows is generic to the types of weeds that I find in my vegetable garden. The bullets are ordered from things that will provide immediate benefit to things that will provide longer-term benefits and/or require more up-front preparation.
Cultivate (shallow hoeing) diligently. If you spend just a few minutes every week with a scuffle hoe you can easily slice off tiny weeds before they become a problem. (Shallow is especially important for plants like cabbage that have shallow root systems. Don't go too deep or you will damage the roots and kill the plant.) Unless you have a very large garden, this is hardly any work at all. Keep your hoe sharp!
Don't let any weeds go to seed. This prevents weeds from growing next season. Taking 10 seconds to pull that yellow dandelion head before it goes to seed saves you 10 minutes of pulling dandelions next year.
Don't till deeply unless you have to. Deep tillage brings seeds from lower in the soil to the surface where they can germinate. If you need to loosen the soil, consider using a fork that does not turn over the soil and bring buried seeds to the surface.
Consider using a propane flame weeder. You have to pay for propane, but this may be less work than hoeing. (I don't use flame, I just use the hoe, so I can't comment much more on the effectiveness here.)
Prepare your beds in a way that prevents weeds from germinating. This is especially possible where you're planting transplants instead of seeding directly. I think it's best to put down mulch before you plant, and then transplant into the mulch. You've got several choices of mulch material, as described in this answer. (Though note that the linked question is about putting down mulch around established plants, so you may find the answers there useful to your situation.)
Also, when preparing beds, consider using the "stale seedbed" method. A couple of weeks before you plant, prepare the bed as if you were going to plant. Water well. Allow weeds to germinate for a week, then cultivate. (Shallowly! You don't want to bring any new seeds to the surface!) Let another batch of seeds germinate for a week, then cultivate right before transplanting. With this technique, you decrease the amount of weed seed that will pop up after your transplants go in.
Consider undersowing a cover crop. You could, for example, sow clover around your established cabbages. The clover is an "intentional weed" that will choke out weeds. Clover is beneficial because it adds nitrogen to the soil. As a bonus, if you have cover crops smothering the weeds, you don't have to hoe!