Where is this information stored?
It's probably under /var/db/sudo or /var/run/sudo and you'll probably find directories of usernames with files under them ordered by tty number.
The actual privileges granted, including how long the sessions lasts before you have to enter your password again depends on how sudoers is setup. There's settings to grant/restrict a lot of different things, but those aren't stored in these files which only store timestamps. How long a session lasts, or when sudo needs to prompt for your password again, is determined by a delta of current time and the session timestamp in this directory, and how long sudo is setup to allow a session to last.
Root is user 0
The key thing is the user ID 0. There are many places in the kernel that check the user ID of the calling process and grant permission to do something only if the user ID is 0.
The user name is irrelevant; the kernel doesn't even know about user names.
Android's permission mechanism is identical at the kernel level but completely different at the application level. Android has a root user (UID 0), just like any other system based on a Linux kernel. Android doesn't have user accounts though, and on most setups doesn't allow the user (as in the human operating and owning the device) to perform actions as the root user. A “rooted” Android is a setup that does allow the device owner/user to perform actions as root.
How setuid works
A setuid executable runs as the user who owns the executable. For example, su is setuid and owned by root, so when any user runs it, the process running su runs as the root user. The job of su is to verify that the user that calls it is allowed to use the root account, to run the specified command (or a shell if no command is specified) if this verification succeeds, and to exit if this verification fails. For example, su might ask the user to prove that they know the root password.
In more detail, a process has three user IDs: the effective UID, which is used for security checks; the real UID, which is used in a few privilege checks but is mainly useful as a backup of the original user ID, and the saved user ID which allows a process to temporarily switch its effective UID to the real user ID and then go back to the former effective UID (this is useful e.g. when a setuid program needs to access a file as the original user). Running a setuid executable sets the effective UID to the owner of executable and retains the real UID.
Running a setuid executable (and similar mechanisms, e.g. setgid) is the only way to elevate the privileges of a process. Pretty much everything else can only decrease the privileges of a process.
Beyond traditional Unix
Until now I described traditional Unix systems. All of this is true on a modern Linux system, but Linux brings several additional complications.
Linux has a capability system. Remember how I said that the kernel has many checks where only processes running as user ID 0 are allowed? In fact, each check gets its own capability (well, not quite, some checks use the same capability). For example, there's a capability for accessing raw network sockets, and another capability for rebooting the system. Each process has a set of capabilities along side its users and groups. The process passes the check if it is running as user 0 or if it has the capability that corresponds to the check. A process that requires a specific privilege can run as a non-root user but with the requisite capability; this limits the impact if the process has a security hole. An executable can be setcap to one or more capabilities: this is similar to setuid, but works on the process's capability set instead of the process's user ID. For example, ping only needs raw network sockets, so it can be setcap CAP_NET_RAW instead of setuid root.
Linux has several security modules, the best known being SELinux. Security modules introduce additional security checks, which can apply even to processes running as root. For example, it's possible (not easy!) to set up SELinux so as to run a process as user ID 0 but with so many restrictions that it can't actually do anything.
Linux has user namespaces. Inside the kernel, a user is in fact not just a user ID, but a pair consisting of a user ID and a namespace. Namespaces form a hierarchy: a child namespace refines permissions within its parent. The all-powerful user is user 0 in the root namespace. User 0 in a namespace has powers only inside that namespace. For example, user 0 in a user namespace can impersonate any user of that namespace; but from the outside all the processes in that namespace run as the same user.
Best Answer
In its most common configuration,
sudoasks for the password of the user runningsudo(as you say, the user corresponding to the process’ real user id). The point ofsudois to grant extra privileges to specific users (as determined by the configuration insudoers), without those users having to provide any other authentication than their own. However,sudodoes check that the user runningsudoreally is who they claim to be, and it does that by asking for their password (or whatever authentication mechanism is set up forsudo, usually using PAM — so this could involve a fingerprint, or two-factor authentication etc.).sudodoesn’t necessarily grant the right to become root, it can grant a variety of privileges. Any user allowed to become root bysudoerscan do so using only their own authentication; but a user not allowed to, can’t (at least, not by usingsudo). This isn’t enforced by Linux itself, but bysudo(and its authentication setup).sudodoes indeed ask for the password after it’s started running; it can’t do otherwise (i.e. it can’t do anything before it starts running). The point ofsudoasking for the password, even though it’s root, is to verify the running user’s identity (in its typical configuration).