When you have a slash chord, the note on the right is the bass note and the note on the left represents the chord to be played. Here's an example...
D/F# = D major chord with an F# as the bass note.
You take your normal D chord (x, x, 0, 2, 3, 2) but switch out the D bass note with an F#, like this (2, x, x, 2, 3, 2).
It's often used in a descending progression for nice voice leading like this - G, D/F#, Em.
** If you have trouble remembering which note is the bass, think of a fraction... the part on the right is the bottom, and the bass is on the bottom (pitch-wise)
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As for your other question, I've done a lot of worship playing (I play for my church, both hymnals for services and choruses for youth), and there are just a few main types of chords that you'll encounter on your chord charts, major, minor, dominant, diminished, and augmented. Extended to 7th chords they are notated Cmaj7, Cm7, C7, Cdim7, and Caug7 respectively. Here are some equivalents as well... Cmaj7 = C?7, Cdim7 = C?7, Caug7 = C+7.
Keep up the hard work, and if you have any other questions, feel free to ask.
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