The very first game, titled 少女たちの真影 (shoujotachi no shin'ei) had a pretty big pun in its title. 真影 (shin'ei) normally means 'portrait,' so the title was typically translated as Portrait of Girls. In fact, however, the word shin'ei is written with the kanji shin, meaning something that is true or genuine, and kage, meaning shadow. Because of that, it can also be interpreted along the lines of "the girls' true shadows," or "girls of the shadows." This is where its official localized title, Skirting Shadows, comes from. Of course, it also gave us the classic 死の美 (shi no bi), pronounced identically to the word shinobi and meaning "the beauty of death."
The second game, Burst, was a bit more straightforward. Its title, 紅蓮の少女たち (guren no shoujotachi) would mean "girls of the crimson lotus," and is where its official localized title, Crimson Girls, comes from. However, it also gave us 紅蓮の焔 (guren no homura), or Crimson Homura. This name can be rather playfully interpreted as referring to red lotuses dancing as if they were flames.
Shinovi Versus was...y'know, a game about shinobi. On the Vita. Shinobi Vita. Shinovi. It...didn't have much else that I can recall. Its official localized subtitle was Proof of Life, though. Because...y'know, boob jokes.
Deep Crimson stepped it back up with the wordplay, though. For starters, its Japanese title, 真紅 (shinku), is a combination of the first kanji from both guren and shin'ei, as this is the game in which the two teams work together. It also gave Asuka an equivalent form to Crimson Homura, known as 真影の飛鳥 (shin'ei no asuka), or Deep Shadow Asuka. This, of course ties, back to shin'ei and guren coming together to become shinku, and like Guren no Homura, Shin'ei no Asuka can be playfully read to mean "a portrait of a bird in flight." If you pair these two together, they also go crazy with the wordplay in their intro, and there's no way I could get to all of it.
Estival Versus is, similar to SV, kind of lacking in the wordplay department. Estival is an actual word, though, meaning something that has to do with summer, so there's that. One of Kagura's intro lines has her say "arata na sekai he izanau," though. Izanau is an antiquated verb meaning to invite someone, and is the root of the names Izanagi and Izanami, the two gods who are said to have created Japan, and after whom Kagura has two attacks named. She also mentions Yamata no Orochi when using her ultimate, Susano'o. Of course, that's not so much a pun as it is just a reference to the legend of the eight-headed snake Orochi and the warrior who slew it, Susano'o.
There is of course, also the name of the series itself, 閃乱カグラ, Senran Kagura. Senran is a play on the word 戦乱 (also senran), meaning war, but the first kanji is replaced with 閃, meaning flash, while the second kanji means rebellion or revolt. 神楽 (kagura) is a form of traditional Shinto dance; the idea of combat being a form of dance is pervasive throughout the series, and this is likely where it comes from. As a whole, though, the series' title is quite a bit harder for a novice like me to interpret. I think it could be taken to mean something like "an instantaneous dance of death," or "a quick, rebellious dance," but I'm not 100% on that.