I think the confusion here arises from the fact that English can use the "-ing" form of a verb in two different ways: using a verb as a noun (gerund), or expressing a continuous action (progressive tense).
In plain language, adding の to a verb in Japanese transforms it into a noun and makes it suitable to be followed by は, が, or various other particles that need to have a noun preceding them:
映画【えいが】を見【み】る。 I will watch a movie.
映画【えいが】を見【み】るの watching a movie (or, "the act of watching a movie")
映画【えいが】を見【み】るのが好【す】きだ。 I like watching movies. (Slightly different translation since the plural is more natural in English.)
映画【えいが】を見【み】るが好【す】きだ。 (incorrect, since が can't follow a verb; it needs a noun.)
So let's throw ～ている into the mix. ～ている shows the continuation of an action, for which English uses the "-ing" form plus a helper verb:
映画【えいが】を見【み】ている。 I am watching a movie.
Translating 見【み】ているの gets a little weird in English, but it can be used like this:
見【み】ているの [the act of] being in the middle of watching
彼【かれ】が映画【えいが】を見【み】ているのを見【み】た。 I saw him [when he was in the middle of] watching a movie.
And of course, remember that ～ている does not always show the continuation of an action in Japanese. It can also show the continuation of a state:
このいすは壊【こわ】れている。 This chair is broken. (not "is breaking")
かばんの中【なか】に携帯【けいたい】が入【はい】っている。 A cell phone is in the bag. (not "is entering")
こと is the other "nominalizer" (read: noun-making machine) that can be used like の to turn a verb into a noun. Sometimes (but not always!) you can interchange the two:
ピアノを弾【ひ】くのが好【す】きだ。 I like playing the piano.
ピアノを弾【ひ】くことが好【す】きだ。 I like playing the piano.
The full differentiation of の and こと as nominalizers is beyond the scope of this question, but as a quick rule, の is generally used when the outer action happens at the same place or time as the inner action, while こと is generally used when the two can be considered from a removed standpoint lacking immediacy. (For further explanation, please see What is the difference between the nominalizers こと and の?.)