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This answer is a community answer, everyone can edit.
Fonts accessible with a simple \usepackage{〈font〉} call:
(*) means that the package makes this font the default text font.
arev - Arev, a sans-serif font (*) with math enhancements; derived from Bitstream Vera Sans
avant - Sets the sans-serif font to Avant Garde
bera - Bera Serif (roman,*), Bera Sans (sans-serif), Bera Mono (typewriter); derived from Bitstream Vera; math font: Computer Modern
bookman - roman font: Bookman (*), sans-serif font: Avant Garde, typewriter font: Courier
cabin - Sets the sans-serif font to Cabin
chancery - roman font: Chancery (*)
charter - roman font: Charter (*)
cmbright - CM Bright, lighter sans-serif variant of Computer Modern Sans Serif (sans-serif, *), CM Typwewriter Light (typewriter)
concrete - Concrete Roman (roman,*) with Euler (math)
courier - typewriter font: Courier
dejavu - DejaVu Serif (roman,*), DejaVu Sans (sans-serif), DejaVu Sans Mono (typewriter); derived from Bitstream Vera
droid - Droid Serif (roman,*), Droid Sans (sans-serif), Droid Sans Mono(typewriter)
ebgaramond - roman font: EB Garamond (*)
euler - Math font
fourier - Sets the serif font to Adobe Utopia (*) and the math font to Fourier (designed to be compatible with Utopia)
fouriernc Sets the serif font to (a clone of) NewCentury Schoolbook (*) and the math font to Fouriernc (designed to be compatible with NewCentury)
helvet - Helvetica
inconsolata - set typewriter font to inconsolata
kpfonts - roman (*), sans-serif and typewriter font from the Kepler Project, text and math symbols
lato - sans-serif font
libertineotf (obsolete) - (Open Type) Linux Libertine (roman,*), Linux Biolinum (sans-serif), and Libertine Mono (typewriter)
libertine - (Type 1 & Open Type) Linux Libertine (roman,*) and Linux Biolinum (sans-serif), and Libertine Mono (typewriter)
librebaskerville - roman font: Libre Baskerville (*)
lmodern - almost identical fonts to Computer Modern (the default TeX fonts), but in Type1 format (outline).
mathdesign - text and maths fonts, versions for Charter, Garamond and Utopia.
mathpazo - changes roman font to Palatino (*), math to something matching.
mathptmx - roman font is Times (*), math is basically Times Italic, with the missing math symbols coming from Computer Modern, RSFS (for \mathcal) and Adobe Symbol
newcent - roman font: New Century Schoolbook (*), sans-serif: Avant Garde, typewriter: Courier
newtx Provides, with the package newtxtext Young Ryu's TXfonts, but with modified metrics (*=Times Roman). Similarly, newtxmath provides the TXfonts math companion fonts with modified metrics. The sans-serif font is based on Helvetica and the typewriter font is especially created.
quattrocento - Quattrocento (roman,*), Quattrocento Sans (sans-serif)
tgadventor Sets the sans-serif font to TeX Gyre Adventor (clone of AvantGarde)
tgbonum Sets the serif font to TeX Gyre Bonum (clone of Bookman,*)
tgchorus Sets the serif font to TeX Gyre Chorus (clone of Zapf Chancery,*)
tgcursor Sets the typewriter type font to TeX Gyre Cursor (clone of Courier)
tgheros Sets the sans-serif font to TeX Gyre Heros (clone of Helvetica)
tgpagella Sets the serif font to TeX Gyre Pagella (clone of Palatino,*)
tgschola Sets the serif font to TeX Gyre Schola (clone of New Century Schoolbook,*)
tgtermes Sets the serif font to TeX Gyre Termes (clone of Times,*)
times (obsolete) - changes roman to Times (*), sans-serif to Helvetica and typewriter to Courier
yfonts - provides Yiannis Haralambous's rendering of Fraktur, Schwabacher and Gotische and a set of beautiful initials.
Fonts not accessible with a simple \usepackage{〈font〉} call:
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