The method is quite old, but still not widely used this day, as people are more inclined to buy some strong stuff - like Roundup - and make sure to kill everything with less effort.  Roundup becomes quite a controversy solution recently, some claim it can cause cancel in people - I guess it depends on a way of using it.

In simple words, I use a gallon of vinegar (5-10%),  200g of salt, a cup of dish soap. Mix the together Then generously spray on leafy weeds in hot sunny day. 

The difference from Roundup  method is:

(a) vinegar does not kill roots, but only leaves or parts of the leaf that it came in contact with, so unlike Roundup which one little drop is enough to kill the whole plant, with vinegar you need to make sure all leaves are covered with the solution;

(b) as soon as you see regrowth from roots you better quickly - before it re-established itself - spray the vinegar solution again - making sure there is no live leaves standing - then roots will eventually die without leaves;

(c) if you accidentally spray on a branch of your "good" plant - this branch will most probably die, but the plant can come up growing again - if no more damage was done. If you try this with Roundup - the whole plant will die, no matter how small the exposure was - it kills the root of the plant, unless you willingly cut out the affected branch immediately to prevent the chemical travel from the leaves down to the root. 

Using more salt in this solution makes it a stronger killer, but caution is needed here as lots of salt - as well as vinegar actually - can change the PH level of your soil. If you don't care about that - then go wild and use as much salt as you want.  And this problem - if ever occurs - can be corrected by adding other ingredients (like limestone) to neutralize the soil again.