Eton is a prestigious British public school for boys. As an aside - in the British education system, a public school is a privately run school that people pay (large) fees to attend - normally a fairly old one. They're called 'public schools' because when they were established, schools were generally owned and operated by groups like the church or trade guilds, and they only educated their own members. Public Schools were open to anybody who could pay. A British public school is equivalent to a private school in the US; schools owned and operated by the government and free to all children are called state schools in the UK, public schools in the US.
Eton College is a particularly well-known and prestigious public school. Both of the current Prince of Wales's children, William and Harry, attended Eton; Nineteen of Britain's Prime Ministers have been Old Etonians; Foreign royal families have been sending their children to Eton for generations. Eton is also one of the most expensive Public Schools in Britain, currently charging about £39,000 per year.
Traditionally, parents would apply for their child's admission at the child's birth - that's no longer required, but it's still common for wealthy or prominent families to register their child early. Registration is commonly referred to as "putting one's name down" - not just in this context, but in any context. One might put their name down for the football team, for volunteer work, etc. It's a contraction of sorts, derived from "Putting (or writing) one's name down on a waiting list"
Justin Finch-Fletchley is saying that he is from a prominent and wealthy family - most likely minor aristocracy, going by his name - and that they had applied to Eton on his behalf when he was born, but he's more excited to be going to Hogwarts.