Which are the most accredited tests for measuring personality traits? I'm lacking of an academic background. I thought there were only 4 or 16 personality traits. But a fast search on google is showing a huge number: according to this link they should be 638.

Is this information correct? 

Which are the most accredited tests for measuring personality traits?
How many personality factors are there?
The history of personality testing can be summarised in terms of an initial period where there was a vast number of personality traits. In response to this, there have been various attempts to synthesise these traits into a smaller number of underlying factors. There is a huge literature on this process (e.g., Digman, 1990; McCrae &amp; John, 1992). 

There are a lot of researchers who have proposed variations on the five factor model (commonly labeled OCEAN: openness, conscientiousness, extraversion agreeableness, and neuroticism). That said, you will find a range of other factorial models (e.g., Eysenk's three factor model; the HEXACO six factor model).

There has also been subsequent challenges to the resulting dominance of factorial models.
For example, see the research of Sampo Paunonen (e.g., Paunonen &amp; Ashton, 2001) which suggests that we may have gone too far in embracing the factorial models of personality. 

Related to this debate, there are a range of hiearchical theories of personality where facets are nested within factors. So you can either get a very general representation of personality with say 5 factors, and a more detailed representation with say 30 facets.

What are valid measures of these factors?
There are a large number of well validated personality tests.
Tests are typically built around particular representations of personality.
There are a range of free measures of the Big 5 listed here with some discussion around validity.

The NEO is one of the most well-known and used commercial measures of the Big 5. That said, there are a wide range of well validated commercial personality tests available.

References


Digman, J. M. (1990). Personality structure: Emergence of the five-factor model. Annual review of psychology, 41(1), 417-440.
McCrae, R. R., &amp; John, O. P. (1992). An introduction to the five‐factor model and its applications. Journal of personality, 60(2), 175-215.
Paunonen, S. V., &amp; Ashton, M. C. (2001). Big five factors and facets and the prediction of behavior. Journal of personality and social psychology, 81(3), 524.