The Claisen condensation is a base-catalyzed reaction between an ester and another carbonyl compound producing a beta-keto ester or beta-diketone as the product.
In a Claisen condensation, just like in other condensation reactions, an enolate attacks the electrophilic carbonyl carbon of a second molecule. What differentiates it from other condensation reactions is that it involves two molecules, at least one of which is an ester.
So, how do we actually form an enolate? We have to deprotonate the alpha-carbon of a carbonyl by using a strong base like sodium ethoxide or sodium amide. Remember that alpha-carbons are much more acidic than standard sp3-hybridized carbons!
Here we have the reaction between a ketone and ester. The stabilized anion (the enolate) attacks the electrophilic carbonyl through a nucleophilic acyl substitution mechanism and the alkoxide is eliminated. If our enolate were a ketone or aldehyde, we'd end up with a beta-diketone (two ketones separated by a carbon); if our enolate were an ester, we'd end up with a beta-keto ester (ketone and ester with a carbon between them).
The reaction between a ketone and aldehyde (or any combination of those) is called an aldol condensation—so named because the nucleophilic addition forms an alcohol.
The Dieckmann condensation is basically an intramolecular Claisen condensation that forms a cyclic beta-keto ester or beta-diketone. In this example, the 2 alpha-hydrogens in blue and green are entirely equivalent since the molecule is symmetrical.