This might be considered too broad to remain open, but I will add some fuel to the fire. These are from memory, and should be reviewed and corrected. I invite others to edit to expand upon my impressions and dim recollections.

Whitehead has the earliest text I am aware of, called Universal Algebra. This was before the modern version of the subject and would contain much found in algebra courses today. Bourbaki has a similar text with a more structured flavor, but again precedes the modern theory. Various advanced algebra texts touch on the modern subject.

Garrett Birkhoff has Lattice Theory, which is essential for understanding much of the work done in Universal Algebra in the last century that involves congruences, varieties, and many other characters of the field.

George Graetzer has one of the earliest comprehensive texts to the modern version of the field. His focus is on partial algebras, but covers much of the initial subject that was known up to the 1960's. I don't know if it has been updated. Graetzer has a modern edition of lattice theory as well, which I have not read.

Burris and Sankappannavar have an introductory book that is part of the Springer series. They are interested in Boolean powers and subjects related to interpretability. However, one can read the first three or so chapters to get a good introduction to the field.

I cut my teeth on Algebras, Lattices, Varieties, Vol. I by McKenzie, McNulty, and Taylor. Chapter 1 gives motivation and an introduction, chapter 2 covers the lattice theory material, chapter three focuses on certain classes of unwary and binary algebras, and chapter four leads to free algebras and the HSP theorem, as well as touches on clone theory and other topics. Chapter five focuses on decomposition into factors and when that is possible. I have seen manuscripts for part of volume II, but nothing in print.

Of course I have been exposed to other topics pioneered by Ralph McKenzie and others. Tame congruence theory and commutator theory are subjects developed in the 80's and 90's, and their titles can be found with a web search. Much has been done about decidability, but I don't know if books have appeared. Search for papers by Jeong, McKenzie, and Valeriote.

Edit: Horrors! Arturo Magidin reminded me of George Bergman's 245 notes, entitled An Invitation To General Algebra And Universal Constructions. I don't know how I forgot that. I must be getting stupid.

Other works have appeared on partial algebras. I believe one such author is Peter Burmeister.

It's possible that Ralph Freese or Bill Lampe have written a book on lattice theory, updating developments there.

Some members of MathOverflow have available texts. Jaroslav Jezek has his version on Universal Algebra, Mark Sapir and Olga Kharlompovich have one concerning algorithmic problems, and some of the aforementioned texts on Model Theory touch on the subject of Universal Algebra. Even some category theorists write on the subject.

I'd be interested in other texts in Universal Algebra published in this century.

Gerhard "Been Out Of The Loop" Paseman, 2019.08.08.

Chapter 2 of Basic Algebra, volume II by Nathan Jacobsonis a really enticing introduction for me. $\endgroup$3more comments