If you're going to get something permanently inked on your skin (or in your skin), you're going to want it to be meaningful. (An eagle wrestling with a lightning bolt while attacking a dragon is meaningful to you? Then by all means.) And all the tattoos that the Dauntless get are pretty meaningful. Tris and Four may have the most meaningful tattoos, but even relatively minor characters like Tori have some symbolic ink.
Tori has a hawk tattoo on her because, she explains, "In some parts of the ancient world, the hawk symbolized the sun. Back when I got this, I figured if I always had the sun on me, I wouldn't be afraid of the dark. […] Now it reminds me of the fear I've overcome." (2.24-6). Here's one reason to love tattoos as symbols: the characters straight up tell you what their ink means. So here, Tori explains that the tattoo has to do with her overcoming her fear of the dark—it's a reminder of something she no longer has.
Tris has two tattoos very much like that. Her first tattoo is of three birds that symbolize her family members. Which is what Tris thinks when she gets them: the three birds are like Tori's tattoo, a "reminder of where she was… a way to honor my old life" (8.116). In case we missed it here, she explains to Four later, that the three ravens are "One for each member of my family" (28.220). So tattoos can be symbols of the past, as in, the kind of past that you wave at and remember as it disappears. This is that tattoo equivalent of writing in a yearbook: "Bye, past, we'll always remember you, have a great summer. KIT."
Another major reason to get tattoos is because tattoos express some part of your identity. (Today, we might just press the "Like" button, but in Tris's day, there's no social media.) Curiously, Tris gets two tattoos because of her identity and they seem pretty contradictory. But hey, that's Tris.
First, she gets the Dauntless seal tattoo with Christina. As Tris notes, tattoos "are a part of life here" with the Dauntless—and here is where she lives (19.29). Let's also throw in the fact that she gets this tattoo with Christina, which is sort of the classic friendship bonding experience (at least, if you were in the army or navy in the 1950s).
So you might see that and say, "Tris is totally accepting her identity as Dauntless." Ah, but not so fast. Just a few chapters later she gets the Abnegation symbol tattooed on her and says, "that symbol is a part of my identity, and it felt important to me that I wear it on my skin" (25.2). Unlike the "these birds represent my family" tattoo, the Abnegation seal symbolizes who Tris is and will always be on some level. She may be Dauntless, but she's also got some Abnegation in her. Which is why these two tattoos help symbolize who Tris is now. And the fact that she gets the Abnegation symbol as a tattoo nicely demonstrates that she's accepting that part of her.
There's one final use for tattoos, which Four nicely demonstrates: it's not the tattoo-as-memory (past) or tattoo-as-identity (present) but the tattoo-as-goal (future). In other words tattoos can symbolize goals. In Four's case, he has a Dauntless seal and the Abnegation seal, just like Tris. But he also has tattoos of the Amity, Candor, and even Erudite symbols (31.74). When he shows these to Tris, it makes for a pretty interesting scene, and not just because he's half-naked.
What's especially interesting is Tris's confusion. So Four has to explain that for him, the tattoos symbolize something of a goal or hope on his part, something both society-related and personal: the tattoos remind him that he wants "to be brave, and selfless, and smart, and kind, and honest" (31.75). These tattoos express his philosophy that splitting up into warring factions wasn't a very smart idea in the first place.