Azurill in depth
Lapis lazuli on legs
The Japanese name, ルリリ (Ruriri), comes from 瑠璃 (ruri), lapis lazuli, the blue stone. English gets to the same place by another route: azure, the colour blue, plus rill, a small stream. Water colour and running water. The “-rill” ending ties it to Marill and Azumarill before you even see the design, which is that of a round mouse with circular ears and pink insides, exactly the same scheme as its evolutions but with the tail taken to the extreme.
It was one of the first Generation III Pokémon to be revealed, on March 7, 2001, alongside Wailmer and Kecleon. There's something amusing about that: the first image the public saw of Hoenn included a baby from a Johto evolutionary line.
The type that arrived twelve years late
Until Generation V, it was pure Normal, without the second half, leaving a member of a Water-type line with not a drop of Water on it despite learning Water Gun at level 1 and Bubble Beam at 6. Generation VI added Fairy, bringing immunity to Dragon, resistance to Fighting and Dark, and the two weaknesses it now carries to Poison and Steel.
That same generation fixed another oddity of its own. Since its gender ratio is three females for every male and Marill's is evenly split, the game recalculated its gender when it evolved, and roughly a third of female Azurill ended up as male Marill. Since Generation VI, gender has been preserved through evolution and the problem disappeared, although it remains the example people cite when talking about that behaviour.
Compared with the other babies
Igglybuff is the natural comparison: another Normal and Fairy baby that evolves with friendship at 160, with 210 base stat points compared with 190 here. They share the same flaw, which is existing to fill out an evolutionary line in reverse, and the same virtue, which is hatching in a handful of steps. The difference is what waits at the end of the road: Azumarill ends up with 420 base stat points, 100 HP and an Attack stat of 50 that Huge Power doubles, while Igglybuff's branch ends up somewhere else.
For a normal playthrough, the recommendation is awkward to write but honest: don't use it. A Pokémon with 20 Attack, 20 Special Attack and 20 Speed doesn't win battles, and the only thing you get from keeping it on the team is the friendship it builds up while being walked around. If you insist on levelling it, the order helps a little — Helping Hand at 3 to soften the blows, Bubble Beam at 6 as its only decent attack for six levels, Charm at 9 to lower the Attack of whatever hits it, and from 12 onwards you can start pushing with Slam and Bounce. At 15, the level-up move list ends, and by then its friendship is usually where it needs to be.
And one last note on the level 1 list, which has three entries: Tail Whip, Water Gun and Splash. One of the three does absolutely nothing.