DCM does not mix with water. You add water solution of NaHCO3 to DCM, shake it, remove water solution (by separation funnel you first remove DCM solution, next water solution, and put DCM back; in a glass, you just pour the top layer off). With this procedure all salts will be broken down to source bases and all water-soluble substances withh be removed from your product. Yes, it's like using brine to dry DCM solution.
Extraction by definition is when you have something in a water layer and you shake it with organic solvent to remove said compound with organic solvent. In washing your product is in organic layer and it stays there, impurities are removed into water layer.
After the reaction is finished ( TLC monitoring),mix the solution with about same volume of saturated solution of sodium carbonate in a separating funnel and shake in order to separate the free desired compound (in the organic layer),reaped other two times. Combine the organic layer ( 3 times) in a beaker and add sodium or magnesium sulfate to absorb the trace of remined water.Finally evaporate the solvent an purify by using flash (Dry) chromatography.
According to your procedure you are trying to react Diamine with 5-(dimethylamino)naphthalene-1-sulfonyl chloride (Dansyl chloride). This reaction will produce a Hydrogen chloride as a by-product in your reaction media which will (or may) react with a free amino group to convert it to aminohydrochloride salt. In that case you need to wash your organic solvent (DCM) with aques Na2CO3 to get rid of the hydrogen chloride and free the amino group.
Washing is just another word for "working up" or extraction that you would do with a separatory funnel. Since you are washing 3 times, I would suggest using 1/3 the aqueous volume compared to your organic layer.
washing or workup both are same words. Extraction it is separation process with Organic solvents (DCm or EtOAc) and aqueous solution by separating funnel. You take aqueous layer 1/4 portion with organic layer repeat the same procedure 2-4 times.
Washing is performed if product is soluble in organic solvent and barely soluble in water and various salt solutions. Organic layer is then preserved along with product, while water is used to remove impurities and other compounds (solvents, acids or bases, etc.).
Extraction is performed if product is pretty soluble in water, but has high affinity for organic solvents. Then aqueous solution of product is mixed with organic solvent (ratio varies), organic layer is removed and preserved, new portion of organic solvent is added. This procedure is repeated until all product does not transfer into the organic layers (monitored via TLC), which are then combined, dried and evaporated dry.
Other types or workup may involve pH manipulation, precipitation, solvent removal, centrifugation and many more- it's a generic term, which means "everything that happens after reaction".