How many types of quarks are there and what are their names?

Like leptons, quarks are divided into three generations.

Each generation contains four quarks- two quarks and two antiquarks.

Each quark is assigned a flavor:
#"up"#, #"down"#, #"charm"#, #"strange"#, #"top"#, and #"bottom"#.

Antiquarks are named respectively- for example, the antiparticle of an #"up"# quark is named #"anti-up"#.

So there are
#3" generations"* 4" quarks"*"generation"^-1=12" quarks"#

Taking the Quantum Chromodynamics (QCD) into account will triple the number. By the QCD model, quarks have three different "colors" (https://tutor.hix.ai)
#"Red"#, #"Green"# and #"Blue"# for quarks, and
#"Anti-Red"# (Cyan), #"Anti-Green"# (Magenta), and #"Anti-blue"# (Yellow) for the respective antiquarks.

So that gives #12" quarks"* 3" color"=36" quark"*"color"#

Sources:
https://tutor.hix.ai https://tutor.hix.ai

Image Attribution:
https://tutor.hix.ai(physics)svg