Sometimes, you might get to the end of a year and be glad it’s over. And it seems that whoever coined the term 忘年会 (bōnenkai) shared that sentiment, as the three kanji individually mean 忘 (bō), forget; 年 (nen, toshi), year; and 会 (kai), party or meet-up.
Together, 忘年会 is used to refer to an “end-of-year party,” and its dictionary definition is “その年の苦労を忘れるために年末に催す宴会” (sono toshi no kurō o wasureru tame ni nenmatsu ni moyo’osu enkai, a party held at the end of the year to forget the hardships of that year.) But more colloquially, it means a chance to get drunk and forget that night. Whether you drink enough to forget an entire year is a different question!
For the past few decades, at least, 忘年会 have been commonly associated with 飲み会 (nomikai, drinking parties) held by whatever company you work at. Typically, depending on the size of the company, this might involve finding a place that does お店まるごと貸切 (o-mise marugoto kashikiri, renting out an entire restaurant). At larger companies, it’s more likely that there will be smaller parties for each 部 (bu, department).