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Questions about slash chords are frequent enough that this question was created to answer the general concerns as well as link to related or more specific questions on the topic.
Chords notated like X/Y: What do they mean, and how does one play them?
The first two bars of "Can't Help Falling in Love" (from here)
The first four bars of "On Green Dolphin Street" (from here)
This example of how to play Queen-style chords on guitar (from here)
For pianists, play the X chord with the right hand and the Y bass note with the left hand. For guitar/bass bands: guitarist plays the X chord and bassist plays the Y bass note. (With thanks to @piiperiReinstateMonica)
There are three main parts to this answer that can be read independently of each other.
Chords of the form X/Y, read X over Y, and sometimes called "slash chords", mean
play chord X, making Y the lowest note.
For example, the first of the "Queen chords", Bb7/D, means play a Bb7 chord, and make the lowest note a D.
Similarly, the second chord in "Can't Help Falling in Love", means play a C major chord, but make the lowest note a Bb. Note that Bb is not part of a C major chord (which contains pitches C E G). This is address below in the "How does one play this?" section.
Chord inversion
When chord X includes note Y, then you're looking at an inversion of X. For example C/E is a first inversion C major chord, played E-G-C (or E-C-G). Similarly, Cm7/Bb is a third inversion Cm7 chord played Bb-C-Eb-G (or Bb-Eb-G-C, etc.)
Passing movement in the bass
Sometimes you want the bass to move smoothly from one chord to the next. Consider the "Can't Help Falling in Love" example in the OP. The basic chord sequence is Bb - C - F - Gm. If taken literally, the bass player would play only the chord roots:
X:0
K:Fmaj
M:4/4
L:1/2
[V:v1 clef=bass] "Bb"B,, "C"C, | "F"F, "Gm"G, |
However, the chord notation is letting the bass player know that a descending step-wise pattern is wanted.
X:0
K:Fmaj
M:4/4
L:1/2
[V:v1 clef=bass] "Bb"B,, "C/Bb"B,, | "F/A"A,, "Gm"G,, |
Pedal tones
Sometimes you want chord to change above a stationary bass pitch, known as a pedal tone The "On Green Dolphin Street" example in the OP demonstrates this. The C is kept in the bass through the entire first four bars while the principal chords change above it.
Notational convenience
Sometimes it's easier for the music reader if the chord notation uses slashes to clarify or simplify things. For example, "On Green Dolphin Street" might have been notated Cmaj9 Cmin7 D7 Dbmaj7. This would be literally correct as far as the notes involved, and an astute player, or one familiar with the tune, would recognize the possibility of the C pedal tone (see #3 above).
However, that's more complex to read than Cmaj9 Eb/C D/C Db/C, which mainly involves triads plus a bass note, and the /C chords make the pedal tone explicit.
In literal terms, this is answered above: you play chord X, placing (or adding) Y as the lowest pitch. In essence, you can play chord X however you want (that is, in any voicing), as long as Y is the lowest pitch.
In practice, this depends on the context you're playing in. If you're a soloist or otherwise responsible for the bass line/lowest pitch, then you make sure Y is the lowest note. But if you are playing with a bass player, or some other instrument responsible for the lowest part, then you probably want to avoid playing Y -- at least not in the same octave as the bass part -- so as not to conflict. (This is a rule of thumb, but not a rule. For more on that, see this question: Should the comping instrument ever double the bass player?)
So, for example, given a G/B chord, if the bass player has the B, then the piano or guitar would play G and D (and possibly B), pitched higher than the bass's B.
In Functional Analysis (Roman Numeral Analysis), slashes are used to indicate a secondary dominant relationship. Secondary dominants are explained in What is a secondary dominant chord?.
There is another form of "slash" chord in which a horizontal line is used, and the notation is written vertically. This denotes a "polychord". See also John Belzaguy's answer to the question Chord Symbols in Kurt Rosenwinkel transcriptions
The 6/9 chord is a pentad with a major triad extended by a sixth and ninth above the root, but no seventh. (Wikipedia)
In this case, the slash is just a separator for readability.
Questions related to slash notation are reasonably common on SE:MP&T. Here's a compendium.
Based on your examples it is almost self explanatory. The X is the chord that is being played and the Y indicates the note that should be played in the bass, lowest pitch note in the chord. That does not have to be played by the bass player but should be the lowest pitch note. As for how to play them, there are several ways to play any chord. Since you have provided chord charts with the specific fingering I'd say play them that way. But without those you might have to work at finding a suitable fingering.
In many cases the slash notation indicates an inversion, for example A/E might be A major second inversion (5th in the bass). In some of your examples the chords are more exotic. You have C/Bb which would be a C7 chord 3rd inversion (7th in the bass). In many cases you can grab a standard chord form and then search for a way to grab the bass note if it's in the vicinity.
Since this question seems very guitar-centric I'd recommend going through Chord Chemistry by Ted Greene, and Mel Bay's Rhythm Guitar Chord System. Two very different approaches but both very useful and valid.