A. Technical methodologies

These are "algorithms" for calculating carbon emissions:

Life Cycle Analysis (LCA) - this is the standard analytical method for calculating the carbon emissions of a product through its lifecycle. OpenLCA is an open source life cycle analysis software.
Product Category Rules (PCR) - these rules define how life cycle analysis should be performed based on the type of product. See this article for one example of the relationship between LCA, PCR, and EPD.

B. Accounting standards

These are standards for calculating and disclosing carbon emissions.

General standards

Environmental Product Declaration (EPD) - comprehensive disclosure of the full environmental impact of a product, where carbon footprint is just one variable. This seems to be used more commonly in building and construction, for example for LEED certifications, but is referenced as a potential source for the Greenhouse Gas Protocol. See an example of a construction material's EPD and an elevator's EPD.
Greenhouse Gas Protocol (GHG Protocol) - this is a comprehensive set of standards for measuring greenhouse gas emissions for a company or a product. It is the base standard for many "carbon neutrality" certifications.
Carbon Disclosure Project (CDP) - these are voluntary disclosures that companies could make through a series of surveys. They are used by large companies to disclose carbon emissions to the investor community.

Industry-specific standards

Whole Building Lifecycle Assessment - specific to buildings and construction, these standards are used to calculate the carbon content of a building based on its construction and materials. This is important because the largest share of a building's carbon footprint is accrued during the construction thereof, rather than the operation thereof. Click here for more about embodied carbon.

Certifications based on accounting standards

These are certifications based on accounting standards, such as the Greenhouse Gas Protocol:

Carbon Neutral - developed and owned by Natural Capital Partners, which has 20 years of experience and implemented this standard for 300 companies in 33 countries.
PAS 2060 - developed by the British Standards Institute (BSI) as an open standard.
Climate Neutral - a new standard developed by entrepreneurs in the fashion industry and currently run by a nonprofit group. Their goal is to make it easier for companies to take climate action and become carbon neutral.

C. Carbon offsets

These are standards for carbon offsets and could in turn be divided into sub-categories:

Mandatory

Mandatory offsets are those based on international treaties, such as the Kyoto Protocol or the Paris Agreement, or by government regulations, such as those in the European Union or California. (I'll leave this category to someone who has more expertise.)

Voluntary

Voluntary offsets are those purchased by companies and individuals to offset their emissions. There are many standards that govern voluntary offsets – some of the best known ones include: