Syllables are the basic unit of sound that you can sound out. This is because they have a peak — something that you can draw out. For example, the one-syllable word <tap> has a peak that you can draw out (the sound of the "a", /&/). This drawn out sound is usually a vowel in English, but some consonants can also take up the role; American English often has no vowel in the last syllable of words like <little> and <rural>, the final /l/ acts as a syllable all by itself.
It's probably of no surprise that people all around the world use the syllable as a fundamental unit in both poetry and song:
There once was an old man of Esser, Whose knowledge grew lesser and lesser, There - once - was - an - old - man - of - Es - ser, (9 syllables) Whose - know - ledge - grew - les - ser - and - les - ser, (also 9 syllables)
Components of a syllable
Syllables aren't just a jumble of sounds that happen to have a peak. As we have already seen, only some phonemes can be the peak in any given language. This is called the nucleus of the syllable. Everything in the syllable that comes before the nucleus is called the onset and everything after it is called the coda. For example, let's take the English one syllable word <once> and determine its nucleus, onset, and coda. First break the word down into its component phonemes.
/w/ /v/ /n/ /s/
Now the nucleus of the syllable must be the vowel, /v/; the onset is everything before that, /w/; and the coda must be /ns/. Syllables don't have to have an onset or a coda, so that the nucleus could be the whole syllable.
Stress and Tone
In most languages, you will find that certain syllables inside a word are "stressed" more than others. This is something you will need to consider for your conlang. English stress is mostly unpredictable, and isn't indicated anywhere in the orthography. When you create your own language, however, you will need either rules of stress that are absolutely regular or a simple way of indicating it when you write your language down.
How should stress be determined? There are many ways you can do that. A very simple way would be to always place stress on a specific syllable: Esperanto, for example, always stresses the penultimate. Another idea is to stress different syllables depending on the phonemes each syllable uses: Latin stresses the penultimate except when both final syllables have short vowels that have no more than a single consonant between them.
The possibilities don't stop there, either. Perhaps stress could be based on the relative "weight" of different phonemes in different places, such as in Spanish: stress the last syllable if that syllable ends with a consonant (except "n", "s"), unless noted otherwise in the orthography. Or you could get rid of the stress distinction altogether, as French has, and instead stress certain words more than others. (French actually stresses the last syllable of phrases, rather than individual words.)
Finally, you could always try using tone instead of stress, but unless your native language is tonal as well you could have a lot of difficulty here. A tonal language distinguishes syllables based on their pitch; for example "high pitch," "extra-low pitch," "low rising pitch" (which starts low before gradually rising), and various others. You can even have "high rising," "rising-falling," or "creaky voice" — in which case the tone is so low the individual vibrations of the vocal chords can be heard. We'd recommend you to examine one or more tonal languages, such as Mandarin Chinese or Vietnamese, and discern the ways those languages make use of it before trying it yourself.
Of course, all of the above refers merely to the stress and tone of individual syllables within a word; if your language isn't tonal, you should also consider the use of stress over entire sentences, though you needn't make it very complicated. English, for example, tends to stress nouns and verbs over prepositions, except when making a point. It also tends to add a slightly higher tone at the end of questions than in statements. (As seen in "This is a sentence" versus "This is a sentence?") You can ignore this sort of thing altogether, or you can make rules of your own.