Any good student of Schoolhouse Rock knows how a bill becomes a law, but how do laws get removed, like obsolete laws?
"The hand that giveth, may taketh away."
Should a legislative body wish to, essentially, delete a law in its entirety, they simply pass another bill to that effect in the same manner as the first - that parity of method is important, anything less is not the hand that gaveth..
These can vary from as little as a single sentence, "Law X is hearby repealed in its entirety," to more comprehensive language, to far more complex versions that only remove part, usually by language such as "Title Whatever, Chapter Whatever, Section Whatever, Paragraph Whatever, of the Such and Such Code is changed to read as follows..." where what follows basically looks like someone edited the existing law with Track Changes turned on.
But in most cases they're not removed, per se. They're frequently just left on the books and go unenforced. Laws that end up in this category are often the subject of entertaining internet articles, but a lack of enforcement is recognized by scholars as having sufficiently similar effect to the law having been repealed.
Laws may also be removed by Judicial action in many jurisdictions, e.g. by being declared unconstitutional or the like, this usually follows the pattern discussed above, creating a condition wherein any attempt to enforce the law will be nullified until the courts (either the same level or higher) decide to nullify the finding of unconstitutionality. In many cases, legislatures or executives will amend such policies, rather than try and wait the courts out. Both of these scenarios, in dizzying array, are found in the history of the U.S.' brief period of banning the death penalty from 1972 to 1976.
An interesting case which was raised elsewhere is known as a de facto repeal, where instead of repealing the specific law in question, some other part of the law is amended or repealed upon which the target law necessarily depended. There are numerous laws on the books that have no effect because the specific cases they name as triggers are now legally impossible to achieve.
Naturally the details will vary from country to country. But the basic way that a law is removed is by being repealed. A government can remove a law by passing a bill.
So in the USA, to remove a Federal law, Congress must pass a bill that "repeals" the law. That bill must be approved by House and Senate, and then signed by the President (like any other bill)
A bill can repeal all, or only specific clauses of a previous act.
That is the main way that a law can be removed. Laws that can't be applied become redundant and void. For example, there are laws regarding "Indians not taxed" in the US law, but since 1924, this category of person doesn't exist, so any law regarding them is void.
Judicial actions can also remove laws, if they are incompatible.
Some laws have specific time limitations, (called "sunset clause") and cease to be laws when their time runs out.
Schoolhouse Rock has misled you.
Congress doesn't pass "laws"; Congress passes bills (so far, so good). The bills passed by the Schoolhouse-Rock process do have the force of law, but are not themselves what you might usually think of as "laws". Instead, think of the whole set of US Laws more like Wikipedia (or any other wiki), and the bills more like individual edits to that wiki.
Most of what you would usually think of as law (as opposed to regulation) is grouped in United States Code. You can read it all here, if you have quite a lot of time on your hands.
If you look at (most) individual pieces of legislation, what they actually do is either appropriate funds, direct an agency to do something, or make changes to US Code (or a combination of all three). The first two probably are not what you think of as laws. The third is usually just a text edit, which gives you the simplest answer: text that can be edited can be re-edited or deleted.
Billy's Law is a clean, recent example. Section 2 is the clear example: (a) directs the Attorney General to go do something. (b) gives him funding to do it. (c) makes changes to a previous law's insertion in US Code.
(the previous change, public law added State requirements for reporting missing children to USC section 5780 of Title 42, which was later moved to section 41308 of Title 34. Fortunately, this was a simple one!)
So: none of the formal legislative rulemaking process is different than Schoolhouse Rock suggests, but the content of a bill is a lot more complex than "School bus must stop at railroad crossing."