When the worlds of Marvel and DC collide they produce some amazing fights. One of the most highly contested battles was between Batman and Captain America in the 1996 crossover comic, Marvel vs DC. Since then they have fought on two more occasions, both in the DC/Marvel movies.
Batman is a formidable fighter able to hold his own in pure hand-to-hand combat against anyone in the universe. He is skilled in multiple fighting disciplines, has a physique and strength far above the average human, and his knowledge of pressure points gives him an edge over even street levelers.
On top of that Batman is a master tactician who can quickly read the actions of his opponents. He can exploit their weak points and come up with counter-attacks that take them by surprise. Batman also has a number of gadgets to help him in combat including tear gas, stun batons, tazers and grappling guns.
On the other hand, Captain America is a super soldier with enhanced strength, speed and agility, lifetimes of military training and a powerful shield that allows him to hurl it hard enough to decapitate tanks or even split a helicopter. He has used his powers against hulks, boxed Thor to a draw, and is a master of all manner of weapons and fighting styles. He is capable of bringing down the strongest heroes in his universe including a fully enraged Hulk. However, if he loses his shield he is vulnerable to the same kind of attacks that any street-level criminal could inflict.