Basically, it means a point between two edges or two places. Left alone, it works with preposition. E.g.: di tengah laut (in the middle of the sea), pada tengah hari (at noon 12:00 PM).
Now, let’s try to add the prefix.
SE-TENGAH which means ‘a half’. E.g.: jam setengah empat (3:30 PM), setengah tahun (half-year), setengah abad (half-century, 50 years). In a repetition ‘setengah-setengah’ like in this sentence: dia bekerja setengah-setengah, it means that he doesn’t complete his work.
ME-(N)ENGAH which means move into the center as a verb; perahu itu menengah menuju samudera (the boat goes to the center of the ocean). While as an adjective, it means the medium category; pekerja kelas menengah (middle class worker).
PE-(N)ENGAH which means one who attempts to bring an agreement between two parties.
The suffixes for this word only exist in one form: TENGAH-AN. It is rarely used; but if you want to say dua tengahan sama dengan satu (two halves equals one), it is still correct. In informal situation, it is common to say pindah tengahan sedikit (move aside slightly to the center).
The word also agrees with both prefixes and suffixes.
ME-(N)ENGAH-I (verb): to bring an agreement for a dispute. Used quite often.
ME+NGE-TENGAH-I (verb): to place oneself to the center of something. Rarely used.
ME+NGE-TENGAH-KAN (verb): synonymous with MENENGAH; bring to the center. To speak out (mengetengahkan pendapatnya = to bring attention to his opinion).
PER-TENGAH-AN (noun): indicating time between the beginning and the end. Also means the average for number.
Hence, for your question: tengah hari (noon), pertengahan tahun (between the beginning and the end of a year), penengah bulan (colloquial way to say pertengahan bulan, never used anymore though), and sekolah menengah (middle school, between elementary and tertiary school). Tadaaa you have learned all forms of the word 'tengah'!
Have fun learning! Hope this helps :)