When you take the official TOEFL IBT test, at a test center or the Home Edition, you get one copy of your score sent to you, and you can have it sent up to 4 institutions at no additional cost (it's included in your registration fee). You can have your score results sent to more institutions at an additional cost. See ETS for more details for how to send scores to institutions and when the institution will receive your scores.
For the IBT, your score will be viewable online in your ETS account in as little as 4 days. You should download the PDF of the score report, and maybe print out a copy of it too. Your scores are valid for two years.
So how is the TOEFL IBT scored? The Reading and Listening sections are marked electronically (by computer). All multiple choice questions are worth one point. But some of the questions are worth more than one point. The directions will always tell you the value of the question if it is worth more than one point. The maximum scaled score for each is 30. How is that done? ETS uses some "magic" (how it's done is kept secret) to create it's scaled scores. ETS does offer a bit of information to help you understand your scores on this page. A high score in Reading is 24-30 and Listening is 22-30.
The Speaking and Writing sections are scored differently. These 2 sections are marked by a combination of AI and trained graders following strict rules to guarantee consistency. Each of the 4 Speaking questions has a maximum value of 4 which is then scaled to a score out of 30. Each of the 2 Writing questions has a maximum value of 5 which is then scaled to a score out of 30. A high score for Speaking is 25-30 and 24-30 for Writing.
If you're doing the TOEFL Essentials test (launched August 2021), which is very different to the IBT, you need to check out the score information at ETS.
This table shows the scaled scores for each of the IBT, PBT and CBT tests:
In the original PBT (the structure of the entire test was nothing like the IBT), you also wrote one essay, called the Test of Written English (TWE). It was scored separately on a scale of 0 to 6. The essay score was NOT part of the PBT score.
Source: Original 2005 ETS PDF, page 5.
There isn't a simple way to convert the scores from one kind of the TOEFL test to another because the structure – types and number of questions – are different. However, ETS does have a table comparing total scores. See pages 6 and 7 of their PDF. On pages 8-10 you can find score comparisons for the Reading, Listening and Writing sections. There's no comparison for Speaking since it only exists in the IBT.
Scores are valid for 2 years. Since the CBT hasn't been offered since 2005, any CBT score will no longer be valid. Any score done with the original PBT is also no longer valid.
For the IBT, ETS originally had (it's from 2007) a PDF file covering score ranges from low to high in each section, giving details on what your skills are and how you can improve your weaknesses. The newer version ETS has, labelled as Performance Descriptors, only explains the skill level demonstrated in each of their score ranges for each of the 4 sections.
If you want an idea of how good your score is compared to others from your country or other countries, ETS does publish detailed reports. A PDF of the detailed report for 2021 can be found here.