Vertical gap between wooden floor and baseboard trim: how do I fix that gap?
Baseboard molding is typically trimmed out against a wood or tile floor using a small molding called a base shoe. Example in the picture below. Due to minor variations in floor height or flooring material thickness over relatively short distances it is not usually feasible to get the wider baseboard molding to push nicely to the floor covering material. The base shoe, being a narrow molding, can flex over distances of a foot or two and fit down snug to the floor. 

Note that when installing the base shoe molding it is nailed into the baseboard and not into the floor. This permits the natural expansion and contraction of the flooring to let it slide in and out from under the base board and base shoe moldings. You would not really want to try sealing the gap under the baseboard down to the flooring material due to the fact that flooring does this expansion and contraction thing.

If you already have the base shoe molding in place the then correct fix is to remove it and then re-install it against the floor. I have found such job much easier to simply replace with new base shoe molding. Base shoe can be pre-finished and then cut and installed. This eliminates the normally less than effective need to try to mask off on both edges if trying to finish in place on the wall. A pneumatic brad nailer is an ideal tool for fastening base shoe into place. 

Base shoe installation requires a miter saw and a coping saw. The coping saw is used when making inside corner joins using a coped joint. 

Note that base shoe is typically not used with carpet installation. The carpet tack strip is typically installed about 3/8" from the baseboard and then the carpet is trimmed to the wall and the edge tucked down behind the wall edge of the tack strip to make a nice clean edge up to the baseboard. Sometimes in older houses carpet is installed over wood floors and in such cases the base shoe is removed before carpeting. New owners sometimes come along and "discover" the older wood floors and decide to rip out the carpet and do a floor sanding and re-finishing. In these cases new base show is the final order of business in trimming out the room.