We know from Joshua 7:15 that the guilty party "and all that he has" will be punished, so Achan and all his family (and their livestock) were killed. This evokes the memory of Korach, leader of the rebellion against Moshe and Aharon; when he and the other rebels were killed (by the earth swallowing them up) their families were also killed (Num 16:33).
But the present verse is unclear; who was stoned, just Achan or also his family, and was anybody instead killed by burning? (We know from the vow that they were killed somehow.)
The end of the verse reads (JPS):
וַיִּרְגְּמוּ אֹתוֹ כָל-יִשְׂרָאֵל, אֶבֶן, וַיִּשְׂרְפוּ אֹתָם בָּאֵשׁ, וַיִּסְקְלוּ אֹתָם בָּאֲבָנִים.
And all Israel stoned him with stones; and they burned them with fire, and stoned them with stones.
One possibility is that there were two stonings: Achan was stoned, the spoils and other property were burned, and then Achan's family (and perhaps the livestock) were stoned. This supports the singular ("him") in the first clause and the plural ("them") in the last one. Rashi takes a similar approach, saying that the latter stoning ("them") refers to the animals. (Rashi does not comment on Achan's family.)
Some other translations, including the (unidentified) one in the question, clarify the timing:
And all Israel stoned him with stones, and burned them with fire, [after] they [had] stoned them with stones. (Judaica Press)
I do not know on what basis these translations make this elaboration. This approach doesn't change the "him", meaning Achan, but suggests that others were stoned either at the same time or immediately after, and then everything was burned.
Arguably the text leaves open the possibility of execution by burning (perhaps Achan was stoned but his family members were burned). This seems unlikely; burning is specified in the text for only two offenses, a man having relations with both his wife and his wife's mother (Lev 20:14) and the daughter of a priest who plays the harlot (Lev 21:9). Neither of these has much to do with Achan's offense. In the biblical text stoning is by far the more prominent method of execution, particularly for public wrongs such as Achan's.
Achan and his family were all stoned. Achan, being the focus of the episode, is called out specifically (they stoned him), and was probably killed first. But we know that he and all his family and all their property were to be destroyed (they burned and stoned them), so in the end everybody died, not just Achan.