In the vowel chart, the rows are the height of the vowel, that is, how high the tongue is in the mouth. The columns are the backness, whether the tongue is toward the front of the mouth or toward the back.
Wikipedia is very helpful with phonetics. There’s an interactive chart on the right. You can click on a sound symbol and see examples in various languages and even hear a recording of the sound:
https://en.wikipedia.org/wiki/Vowel
In the consonant chart, the columns are the place of articulation, i.e. where the sound is produced and the rows are the manner of articulation, i.e. how the sound is produced.
Rather than post the maximal, highly complex Wikipedia consonant table, here’s a link to the consonants found in English:
https://en.wikipedia.org/wiki/English_phonology#Consonants
You can click on the sounds to read/hear examples.
@Mimishu1995‘s explanation is good, just wanted to make one correction. What you’re describing as “approximant” is actually an “affricate”. Affricates are two sounds, a plosive followed by a fricative, like the “ch” in “church” or the “j” in “jar”. Approximants are sounds where the parts of the mouth barely touch each other, like the “y” in “yarn” or the “w” in “war”.