A comprehensive catalog of 623 galactic eclipsing binary (EB) systems with eccentric orbits is presented with more than 2830 times of minima determined from the archived photometric data by various sky-survey projects and new photometric measurements. The systems are divided into two groups according to whether the individual system has a GCVS name or not. All the systems in both groups are further classified into three categories (D, A, and A+III) on the basis of their eclipse timing diagrams: 453 D systems showing just constantly displaced secondary minima, 139 A systems displaying only apsidal motion (AM), and 31 A+III systems exhibiting both AM and light-time effects. AM parameters for 170 systems (A and A+III systems) are consistently calculated and cataloged with basic information for all systems. Some important statistics for the AM parameters are discussed and compared with those derived for the eccentric EB systems in the Large and Small Magellanic Clouds.