From the Oxford:

rhyme |rīm|
noun
correspondence of sound between words or the endings of words, esp. when these are used at the ends of lines of poetry.
• a short poem in which the sound of the word or syllable at the end of each line corresponds with that at the end of another.
• poetry or verse marked by such correspondence of sound.
• a word that has the same sound as another.

To simplify:

rhyme, the correspondence of sounds between words or the ending of words.
rhythm is a regular, repeated pattern, from the Greek rhuthmos, ‘to flow.’

To review:
syllable, a vowel sound usually conjoined to a consonant.
vowel, an open vocal-tract sound.
consonant, a closed vocal-tract sound.

Now, some examples. (Note that rhyme at the end of the line is only one of many types):

triple, words with three rhyming syllables, e.g., quickening/thickening
head, i.e., alliteration: “And the silken sad uncertain rustling of each purple curtain.”
end, rhyme at the line’s end
apocopated, a “cut-off”rhyme, e.g., hot/potted, pain/gainless
internal, rhyme within the line
rising, an iamb (syllables unstressed and stressed) or single stressed syllable at line’s end (see image below)
slant/half/off/approximate/near, an “almost” rhyme, e.g, fear/care, gone/moan
homonym, a repeated rhyme with different spelling, e.g., sail/sale, preys/praise
identical, or repetition, the same word, reemphasized
pure, rhyme with differing initial consonant: bell/cell/dell/fell/hell
falling, a trochee (syllables stressed and unstressed)(see image below.)
unpatterned, randomly-placed rhymes
eye, slant rhymes that look alike: cough/rough, wind/find
linked, end syllable of one line beginning the next, e.g.,
Night weighs down on the rooftop
stops the flashlight of a scared cop

The list is an adaptation of from
Mayes, Frances. The Discovery of Poetry. Harcourt. 2001.

The stressed or accented syllable is indicated by the ictus′ and the unstressed or unaccented syllable by the breve˘