Factor analysis, in which both latent (unobserved) and manifest (observed) variables are continuous, is perhaps the best known.

In latent profile analysis the latent variable (e.g. consumer segments) is categorical and the manifest variables (e.g. responses to rating scales) are continuous.

Latent trait models (e.g. item response theory) are characterized by continuous latent variables and categorical manifest variables (e.g. correct or incorrect answers to test items).

In latent class analysis both latent and observed variables are categorical.

There are also hybrid models which include both continuous and categorical latent and manifest variables.

In some models there is a distinction between dependent and independent variables. Censored, truncated and count variables can also be accommodated.

Any of these models can be multilevel (hierarchical) or longitudinal and can incorporate exogenous variables (covariates).

This popular book is focused on latent class analysis and its longitudinal extension, latent transition analysis. It is well written and covers theoretical and technical issues as well as application.

https://www.google.com/search?kgmid=/g/12bmhby6b&hl=en-JP&kgs=a09137cca2d41ecf&q=Latent+Class+and+Latent+Transition+Analysis:+With+Applications+in+the+Social,+Behavioral,+and+Health+Sciences&shndl=0&source=sh/x/kp/osrp&entrypoint=sh/x/kp/osrp

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