What is a D-brane? I know that in string theory, D-branes are objects on which open strings are attached with Dirichlet boundary conditions. But what exactly is a brane? Are they equally fundamental objects like string? If so then do they also vibrate? If the visible universe itself is not a brane then what is the dynamics of these branes within the universe? Do individual D-Branes interact, collide? Can an open string tear itself off from the D-brane? If so what are the results?
 A: Branes are (usually) extended objects; $p$-branes are objects with $p$ spatial dimensions.
D-branes are a special and important subset of branes defined by the condition that fundamental strings can end on the D-branes. This is literally the technical definition of D-branes and it turns out that this simple fact determines all of the properties of D-branes.
Perturbatively, fundamental strings are more fundamental than branes or any other objects. In that old-fashioned description, D-branes are "solitons" - configurations of classical fields that arise from the closed strings. They are analogous to magnetic monopoles - which may also be written as classical configurations of the "more fundamental fields" in field theory. In a similar way, D-branes' masses diverge for $g\to 0$.
Non-perturbatively, D-branes and other branes are equally fundamental as strings. In fact, when $g$ is sent to infinity, some D-branes may become the lightest objects - usually strings of a dual (S-dual) theory. When we include very strongly coupled regimes (high values of the string coupling constant $g$), there is a brane democracy.
Back to the perturbative realm. The condition that open strings can end on D-branes - and nowhere else - means that there exists a particular spectrum of open strings stretched between such D-branes. By quantizing these open strings, we obtain all the fields that propagate along (and in between) such D-branes. The usual methods (world sheets of all topologies, now allowing boundaries) allow us to calculate all the interactions of these modes, too.
So yes, D-branes also vibrate. But because their tension goes to infinity for $g\to 0$, you need even more energy to excite these vibrations than for strings. The quanta of these vibrations are particles identified with open strings - that move along these D-branes but are stuck on them. The insight that the D-branes are dynamical and may vibrate, and the insight that they carry Ramond-Ramond charges (generalizations of the electromagnetic field one obtains from superstrings whose all RNS fermionic fields are periodic on the world sheet) were the main insights of Joe Polchinski in 1995 that made D-branes essential players and helped to drive the second superstring revolution.
Other branes typically have qualitatively similar properties as D-branes but one must use different methods to determine these properties.
When we quantize a D-brane, we find open string states which are scalars corresponding to the transverse positions. It follows that D-branes may be embedded into the spacetime - in any way. The shape oscillates according to a generalized wave equation again. Also, all D-branes carry electromagnetic fields $F_{\mu\nu}$ in them. These fields are excited by the endpoints of the open strings that behave as quarks (or antiquarks). For a stack of $N$ coincident branes, the gauge group gets promoted to $U(N)$. The electric flux inside the D-branes may be viewed as a "fuzzy" continuation of the open strings that completes them to "de facto closed strings".
Those fields have superpartners in the case of the supersymmetric D-branes which are stable and the most important ones, of course. D-branes may collide and interact much like all other objects.
The most appropriate interaction that allows the open strings to "disconnect" from D-branes is the event in which two end points (of the opposite type, if the open strings are oriented) collide. Much like a quark and antiquark, these two endpoints may annihilate. In this process, an open string may become a closed string - which may escape away from the D-brane. The same local process of "annihilation of the endpoints" may also merge two open strings into one. Such interactions are the elementary explanations of all the interactions between the fields produced by the open strings - for example between the transverse scalars and the electromagnetic fields within the D-brane.
Aside from that, some branes may also be open branes, and end on another kind of branes. The latter brane always includes some generalized electromagnetic fields that are sourced by the endpoints or end curves or whatever is the $(p-1)$-dimensional geometry of the boundary of the former brane.
