Scholars and scientists often want to know what impact they are having on their field of study. The most traditional way of quantifying this is to determine how many times one's publications have been cited in the literature. Citation indexes make this fairly easy to do, but to ensure accuracy it has to be done correctly and consistently.
The h-index is a measure of "citedness" as a proxy for productivity and impact. It is the number of articles h in a group of publications N that have received h or more citations. For example, an h-index of 20 means that there are 20 items in the selected group N that have received 20 or more citations. It is like a median, and useful because it discounts the disproportionate weight of highly cited and uncited papers that would skew a mean.
However, the h-index is not an absolute number and will vary considerably depending on several factors. For example, different databases (e.g. Web of Science vs Google Scholar) cover different segments of the literature and use different methods to count citations.
Comparing h-indexes: An author's number of publications and the length of time they've been active affects the h-index. Older and more prolific authors will usually have higher h-indexes than younger or less prolific authors. If you want to compare your h-index to someone else's, you need to use the same methodology to calculate them and then normalize the values by dividing them by a second factor, e.g. years since PhD. The standard caveats apply when using h-indexes in personnel and funding decisions.
Web of Science is generally considered to be best at capturing metrics from quality sources only due to its selective indexing of largely "high influence" journals and proceedings - meaning that it represents what is typically considered to be the most robust and highest quality publications (though any potential authors should do their own analysis of a journal's quality before publishing in it instead of trusting that because it has made it into one of these indexes it is certainly reputable). This means that it can miss some publications which do not fit in its scope. This can be things like less well known journals - either because they are very niche or due to newness - as well as nonstandard types of literature such as 'grey literature,' white papers, and other types of writings not published in traditional high impact journals. It also does not index as many books as the other options (google scholar indexes the most). As a result, it is considered more robust for STEM fields than for arts, humanities, and social sciences because many of the latter groups' fields use more non-journal publications in determining impact metrics - though it can still be useful for those fields, so long as it is used with the knowledge of what might be excluded from the metrics.
Things to note when performing an author search in Web of Science:
How to perform an author search:

Scopus is a product of Elsevier, and like Web of Science, is a curated index - all journals and publishers selected to be indexed are chosen by an independent content selection and advisory board. It is similar in size to Web of Science, though it collects significantly more books.
Things to note when performing an author search in Scopus:
How to perform an author search:

Google Scholar has the largest index of the three, and indexes the widest range of types of publications - including preprints, proceedings, white papers, some grey literature sources, everything found in google books, and more. Because of this, google scholar metrics will have more citations and links between types of publications across subject areas - but it comes with some caveats as well.
Citation metrics are only as reliable as the underlying data. Google Scholar's metrics are generally not reproducible and will differ - sometimes significantly - from data found in Web of Science. Google indexes a different, wider (and largely unknowable) universe of publications. It is also very difficult to resolve author name ambiguity.
Depending on the type of citation information you need, google scholar can be an excellent choice - but be sure to be aware of the downsides as well!
Things to note when performing an author search in Google Scholar:
How to perform an author search:

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