You (almost) never choose your own name. Names were very political and your lord/father/someone important did. Note that all of the below only apply to men of some social standing (usually).

The 幼名 (yōmyō youth name) is the one received at birth. For boys, names ending in 千代 (chiyo meaning "thousand year/generations/forever") or in 丸 (maru meaning "all" though there are various theories on what it originally meant when used in a name) are common. One common ways of naming is to pass down your own yōmyō to your heir. 竹千代 (Takechiyo "thousand generation bamboo", though take can have many different meanings if taken as a homonym) for instance was passed down from Tokugawa Ieyasu to many of the Tokugawa Shōguns. The other one is to pick some auspicious word for the name. Takeda Shingen's yōmyō was supposedly 勝千代 (Katsuchiyo, "victory") because he was born when his dad won a battle. The name passed down in the Hosokawa clan was 聡明丸 (Sōmeimaru, "smart/wise"). Toyotomi Hideyoshi named his first son 棄丸 (Sutemaru "abandoned") on the common believe that abandoned children that were later taken in tended to grow up strong and well. Nature ended up not caring for Japanese superstition and the child died when he was two, so Hideyoshi named his second son 拾丸 (Hiromaru, "found") who grew up to become Hideyori.

On reaching adulthood you received two names.

First, your first 仮名 (Kemyō, "fake/temporary name") or 通称 (Tsūshō), which is what people called you. Most commonly this would be the 輩行名 (Haikōmei), which signifies the order of your birth: Tarō for the eldest son, Jirō for the second, Saburō for the third and so on. Again, this is not set in stone. Tokugawa Ieyasu was Jirōsaburō despite being the eldest son (it seems he inherited from his father according to Edo sources, though only Saburō could be confirmed from surviving documents). Ii Naotora, the famous female leader of the Ii valley, was given (or took) Jirō, the same name as his father, as she was the only child and circumstances forced her to inherit the clan. Sanada Nobuyuki, the founder of Matsushiro Domain, was Genzaburō (third son of the Minamoto, or "Genji", the Sanada say they're descended from the Seiwa Genji line) despite being the eldest, apparently on the belief that the eldest son had a tendency to die young. This is what close family members/friends/relatives would call you for the rest of your life, even after picking up other 仮名.

These other 仮名 were basically titles, like 内府 (Naifu for Naidaijin "interior minister") or something-something-no-守 (Kami, "lord of..."), or something-something-兵衛 or 衛門 (bei or emon, "guard...") or something-something-no-介 (suke, "vice minister..."). Obviously these changed as you get promoted, and what title people called you changed with it (after all, you wouldn't call lieutenant Dan lieutenant once he's promoted to captain). Originally these were formal titles appointed by the Kyōto court. But by the Sengoku many lords were handing out these titles to match the domain's positions or as prestigious rewards, and often the title had absolutely no relationship to what the person's job was, the most often case being people being titled the lord of some far off province they've never set foot in and the domain had no control over. While the court still officially assigned titles, so someone got to be the official lord, people also gave themselves titles for prestige. For instance, 新免武蔵守 ("Shinmei, the Lord of Musashi"), aka Miyamoto Musashi, was nowhere near Musashi province and was not the official holder of the title. These titles would be how most people would (or should) address you.

Your other personal name, likely the personal name you're thinking of, is the 諱 (Imina, "forbidden name"). As the name implies, it is consider rude to address someone by their 諱, unless you're just stating their full name. You'd only address someone by only their 諱 if they're your enemy. 諱 are usually two characters. One is usually something passed down in the family, like the 信 (Nobu "trust") of the Takeda and Oda, or the 景 (Kage "shadow" or "scenery") of the Nagao, or the 元 (Moto "origin") of the Mōri. In the Edo period it seem to have became just pick one character from the previous lord instead of having the same character throughout multiple generations.

The other would be one from someone important. Your lord or your eboshioya (the person in charge of your coming-of-age ceremony) were common choices and they were often the same person. Tokugawa Ieyasu originally got 元 from his lord Imagawa Yoshimoto (義元). Getting a character from someone in the Bakufu is also a show of prestigue, like how Takeda Harunobu (Shingen) got the 晴 (Haru, "spring" or "sunny") from the Shōgun Ashikaga Yoshiharu. Though this is not set in stone. For instance we don't know where the 長 (Naga "long") in Nobunaga came from, though he gave it to many of his vassals.

The two characters were, unsurprisingly, political. One signifies your tie to your father or clan (birth or adopted). The other signifies your tie to your lord. And both could change as circumstances changed. When Matsudaira Motoyasu declared independence from the Imagawa, he got rid of the 元 and became Ieyasu. During the Sekigahara campaign, Sanada Nobuyuki had sided with Ieyasu while his father Masayuki sided with Ishida Mitsunari. The character 幸 (Yuki "luck") was one passed down in the Sanada clan, but after the campaign Nobuyuki took the character 之 (also pronounced Yuki) to show him severing his tie with his father.

You can read here for some more information on Japanese naming convention, by /u/cckerberos.