Only if command1 is successful (exits with exit code 0), will command2 get executed.

If command1 fails, then command3 will get executed.

If command1 is successful, and command2 fails, then also command3 will get executed.

This last example demonstrates the difference from a classic if statement:

if command1; then command2; else command3; fi

With this if statement, if command1 is successful, and command2 fails, command3 will not get executed, contrary to the ... && ... || ... form.

Consider this:

ok() { echo $1; }
fail() { echo $1; return 1; }

echo case ok ok
ok 1 && ok 2 || ok 3
# -> outputs: 1 2

echo case ok fail
ok 1 && fail 2 || ok 3
# -> outputs: 1 2 3

echo case fail ok
fail 1 && ok 2 || ok 3
# -> outputs: 1 3

echo case fail fail
fail 1 && fail 2 || ok 3
# -> outputs: 1 3

echo case ok ok
if ok 1; then ok 2; else ok 3; fi
# -> outputs: 1 2

echo case ok fail
if ok 1; then fail 2; else ok 3; fi
# -> outputs: 1 2

echo case fail ok
if fail 1; then ok 2; else ok 3; fi
# -> outputs: 1 3

echo case fail fail
if fail 1; then fail 2; else ok 3; fi
# -> outputs: 1 3