SPQR is an initialism from a Latin phrase, Senatus Populusque Romanus, which was the motto of the Roman Empire and translates to "The Senate and People of Rome". It links together the original struggles between the senators, the people of Rome and the early Roman Empire.
It was forged during the Republic but the emperors kept it, and it can still be seen on some monuments in Rome. The campers of Camp Jupiter also use this motto in their everyday living as well as their tattoos.
All centurions are seen with the SPQR tattoo along with their godly parent's symbol and at least one bar line. Reyna Avila Ramírez-Arellano's tattoo consists of the initials SPQR, a crossed sword and torch, the symbol of her mother, Bellona and 4 bar lines, representing her four years of service. Hazel Levesque's consist of the same initials, a hieroglyph-like symbol of Pluto, and one bar line. At the council/principia before their quest, Frank Zhang receives his tattoo with Mars' symbol, the crossed spears, and one line. Near the end of the book, Percy Jackson also receives his tattoo when he takes Jason Grace's place as a praetor of Camp Jupiter. His tattoo consisted of a trident, Neptune's symbol, and a line for his first year of service in the legion.
When a camper arrives at Camp Jupiter they receive a probatio tablet. Once they have proven themselves to the legion, they will receive the SPQR tattoo. This tattoo is deeply burnt into their arm from the heavens. The tattoo includes vertical lines symbolizing their years of service and the SPQR tattoo. They also have the symbol of their parent (unless you're a legacy, like Octavian).
This tattoo proves and signifies that you are a full member of the legion. It is impossible to remove this tattoo, as it is engraved in your skin and it is presumably very painful to get.