Vaccines are given to the public to prevent serious disease. Usually the risk of serious disease is small, therefore the risk for serious side effects of the vaccine given must be very small.
Since vaccines are given to a large part of the population, the claims for damages and treatment of side effects (even though rare) can become very costly. It is customary that countries arrange compensation programs to handle such costs. In this way the manufacturer is protected against such costs that other wise would jeopardize the profitability of the product.
Approval
Vaccines are closed as a biological product and actually not as a drug even though often referred to as such. One difference hereof is that vaccines do not need to go through same safety studies as drugs.
Authorities and manufacturers have several different ways of getting new vaccines approved. Here we mention pharmaceutical testing, fast track, animal rule and emergency authorization.
Pharmaceutical Testing
Development of vaccines are divided up in several different stages. The products relative security, purity and potency is evaluated in every step before you go on to the next one to avoid that humans will get hurt inadvertently.
During the preclinical part (approximately 2 years) the experimental vaccine is given to animals to evaluate its safety and potency to prevent disease.
During the clinical part, the vaccine is given to humans consists of 3 phases.
During the first phase the vaccine is given to a small group young and healthy individuals and the products ability to activate antibody production is measured and possible side effects are mapped out.
In phase 2, the vaccine is given to a larger group (several hundred) healthy adults to further affirm safety and efficacy of the product and also to control for side effects. Another part of phase 2 is to decide the optimal dosage of the vaccine. For practical reasons the approved vaccine will be given to everyone in the same dose, none dependent on body weight which is usually the case with other drugs.
In phase 3 (5–10 years) studies are made in large scale on thousands of people to determine the efficacy of preventing disease and also more rare and long-term side effects.
Finally there is a waiting period of up to to years to get the approval of FDA (American Food and Drug Administration) or EMA (European Medicine Agency).
Fast Track
To be able to more quickly market new vaccines the manufacturer can apply for lower demands for security and potency. When fast track is granted, you do not have to show that the vaccine has a direct effect on preventing disease. It is enough that there is a probability that it can prevent disease.
Fast track has been allowed when there is delivery problems of the already approved vaccine.
Animal Rule
When examination of the vaccines effect on humans is deemed unethical or impossible it can still get approved. In that case you use the animal rule when no human studies are made. If the vaccine has effect on animals it is deemed probable to work on humans.
Emergency Authorization
Exceptionally, the demands for potency and safety can be lowered even further. This can happen if a licensing authority declares a state of emergency and makes the judgement that there is a risk that many people can fall sick. In a state of emergency non approved vaccines can be allowed to be used.

Covid-19 Vaccines
None of the Covid 19 vaccines that are used today are approved. They have only been given state of emergency authorization and none of the clinical studies are finished.
Source: Biopharmaceutical, Wikipedia, https://en.wikipedia.org/wiki/Biopharmaceutical
Source: Biological medicinal products, EMA, 2016, https://www.ema.europa.eu/en/documents/leaflet/biological-medicinal-products_en.pdf
Source: Regulation and testing of vaccines, Baylor och Marshall, Elsevier Public Health Emergency Collection, 2013, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7152312/
Source: The National Childhood Vaccine Act, 1986, https://dash.harvard.edu/handle/1/8852138
Source: Comparison of COVID-19 Vaccine Approvals at the US Food and Drug Administration, European Medicines Agency, and Health Canada, Lythgoe och Middleton, JAMA, 2021, https://jamanetwork.com/journals/jamanetworkopen/fullarticle/2781352