Citation analysis, or bibliometrics, is a method to gauge the impact of specific works (journal articles, books etc) on the field, based on the frequency, patterns, and graphs of citations in documents.
It can be simply about citation counts to papers or as complex as tracking citation collaboration networks between authors and organizations.
Several databases and sources can be used to harvest citation data and depending on from what source you access it, you will likely find different citation counts.
Currently, there are four major bibliographic data sources (Web of Science, Scopus, Dimensions and Google Scholar) with their overlap in the millions of records (with an estimate for Google Scholar at 389 million records in a 2018 study).
The Web of Science (WoS) Core Collection consists of 89 million records and is used mainly by researchers and administrators to analyze research impact.
It is also the database used for Journal Citation Reports (JCR), which is where the journal metrics including Journal Impact Factor are published.
Read more: Citation analysis and databases containing citation information / Oulu University Library
H-index is also known as the Hirsch index and Hirsch number. It's developed in 2005 by Jorge E. Hirsch.
H-index is a computable index to estimate the importance, significance and broad impact of a scientist's cumulative research contributions.
Author's H-index means that an author has published h papers each of which has been cited (by others) at least h times.
The value of h-index is calculated by scientific field (different publishing and citation practices), length of researcher's career (longer time-window enables more publications and citations) and by total number of publications (h-index can't be larger than number of publications). Author's h-index can be found in Web of Science, Scopus and Google Scholar.
Check out this article on how to calculate H-index.
Contemporary h-index (hc-index), by Sidiropoulos, Katsaros & Manolopoulos (2006) gives more weight to recent and highly cited research than to older research by adding an age-related weighting to each cited article, giving less weight to older articles regardless they are highly cited.
M-index is calculated by dividing the h-index with the number of years that an author has been active, defined as the years since the date of first publication.
It corrects the fact that the h-index is influenced by the age and length of career of the researcher. It provides a comparison between researchers within a field at different career stages. However it has an assumption of a continuous research activity since the author's first publication.
i10-index was created by Google Scholar and is used in Scholar's "My Citations" feature. It is straightforward to calculate and free to use. i10-index = the number of publications with at least 10 citations
Check out this article on how to calculate i10-Index.
G-index was proposed by Leo Egghe in his 2006 paper "Theory and Practice of the G-Index" as an improvement on the H-Index.
It is calculated by: "[Given a set of articles] ranked in decreasing order of the number of citations that they received, the G-Index is the (unique) largest number such that the top g articles received (together) at least g^2 citations." (from Harzig's Publish or Perish Manual). It highlights highly cited articles of authors.
More about G-Index in this article.