The result of all of this infrastructure is that the Linux kernel runs in a set of Linux processes just as it does on physical hardware. The machine-independent portions of the kernel can't tell that anything strange is happening. As far as they are concerned, they are running in a perfectly normal machine.
When user-mode Linux is started, the normal kernel boot-up messages are written to its console, which is the window in which it was run. When the kernel has initialized itself, it runs init from the filesystem that it's booting from. What happens from that point is decided by the distribution that was installed in that filesystem.
Essentially all applications that run on the native kernel will run in a virtual machine in the same way. Examples include all of the normal daemons and services, including the Apache web server, sendmail, named, all of the network services, and X, both displaying as a client on the host X server, and as a local X server.