It's in the name! Carboxylic acids are somewhat acidic having pKa in the range of 4-5 for typical carboxylic acids.
The acidity of carboxylic acids can vary depending upon their structure and substituents attached to them. In general, the more the electron-withdrawing ability of the groups attached to the carboxylic acid the more acidic it is. For example, the following pKa trend illustrates that more electroneagtive halogens or more of them results in a more acidic acid.
Recall that a halogen is an inductive withdrawer so distance is important. The closer the halogen is to the carboxyl the more acidic it is.
This is most easily rationalized by inspecting the acid/base equilibrium and the stability of the conjugate base. If the conjugate base is very stable then the equilibrium will lie further to the right, producing more H3O+ (i.e. more acidic). Increasing the electron-withdrawing nature of the R group, stabilizes the conjugate base (carboxylate ion), pulling the equilibrium to the right.
This is also observed in substituted benzoic acids. For example, have a look at the following trend.
The more electron-withdrawing the substituent the more acidic the benzoic acid (e.g. NO2 is a powerful electron-withdrawing group). The electron-donating -OH groups destabilizes the conjugate base relative to that for unsubstituted benzoic acid so it is less acidic.
As we've seen above when a carboxylic acid is dissolved in water, the acid undergoes dissociation producing a carboxylate (RCO2-) and a proton (as H3O+). Under these circumstances typically only a small amount of the carboxylate is formed. If a carboxylic acid has a pKa of 4 to 5, then the Ka equilibrium constant is on the order of 10-4 to 10-5. Therefore there is a small amount of carboxylate formed.
However if we treat the carboxylic acid with a base that has a conjugate acid with a pKa greater than 4 to 5, then the carboxylate will form.
Treating a carboxylate with a proton source (e.g. HCl) protonates the carboxylate and yields the carboxylic acid back.
A carboxylate is a salt (i.e. ionic compound) and is therefore highly soluble in water. Carboxylic acids with R groups greater than 4 carbons are typically insoluble in water and soluble in organic solvents. Organic chemists take advantage of this when isolating carboxylic acids from other compounds. Dissolve your mixture in an organic solvent, then extract with NaOH (aqueous). the carboxylic acid will convert to the carboxylate and dissolve into the aqueous layer. Separate the layers and then acidify the aqueous layer and the carboxylic will precipitate (crash out).