Where did the practice of not counting people come from? People count not-one, not-two, not-three etc. What is wrong with counting one, two, three etc? This is often found when counting to see if there is a minyan in shul.
In the Torah (Ex. 30:11), Hashem commands Moshe to count the Jewish People via half-shekel donations, "so that there not be a plague when they are counted." Later, too, we find David conducting a direct census, and a plague breaking out (II Sam. 24:1ff). The exact cause-and-effect mechanism is variously explained by the commentaries, but at any rate, we see that there is reason to use some kind of proxy for counting Jews.
King Saul, we are told, used first stones (I Sam. 11:8) and later sheep (ibid. 15:4) for his censuses - both as explained by the Gemara (Yoma 22b). In the Temple, when they had to count the Kohanim to decide who would perform the various services, they had each of them stick out a forefinger (ibid.; Rambam, Laws of Daily and Additional Offerings 4:3).
The Rambam rules (ibid. 4:4) based on the above that indeed it is halachically prohibited to count Jews directly.
A common practice, when counting for a minyan, is to use the verse "Hoshiah es amecha..." (Psalms 28:9), which has exactly ten words; this is cited in Kitzur Shulchan Aruch 15:3. In one of the Lubavitcher Rebbe's (o.b.m.) talks (published in Shaarei Halachah Uminhag, vol. 1, pp. 237-238 - available online here), he points out that using this verse (in contrast to an older custom, recorded by Rashi, of using instead Psalms 5:8) has the additional benefit of expressing our most urgent wish for the salvation and redemption of the Jewish People by Moshiach.
this is specifically for jews. when we were in the desert the census were made with rocks. instead of counting the people we would do it indirectly, each person would put a small rock in a pile and then we count the rocks.
david hamelech counted the people for the census and we got a kelala.
and alternative (and funnier) way to count people for a minian is: "baruch ata hashem elokenu melech halolam hamosi lehem min haares"