Lots of transgender people feel something called "gender dysphoria," which is the discomfort or distress of varying intensities with one's body, physical attributes, behaviors, etc., due to the incongruence of assigned sex and gender identity. Trans people experience dysphoria in all kinds of ways. There is no one way to experience dysphoria.
A transition includes any step a person takes to affirm their gender identity. Hence, there is no one way to transition. A transition can include a haircut, hormones, surgery, a wardrobe change. But it can also not. Many people break acts of transition into two main categories:
Social transitioning can involve things like telling people that you'd prefer different pronouns or a different name. It can involve a wardrobe change or a change in how one socializes. But it does not have to. For me, social transitioning meant asking people to refer to me with he/him/his pronouns and shopping in the men's section for clothing again. Social transition involved little to no change in my social behaviors.
Hormone Replacement Therapy: this entails taking testosterone (commonly abbreviated as "T") in various forms (see below) or estrogen. These cause an individual to go through either testosterone driven puberty (wherein the voice drops, red blood cell count increases, body hair growth increases, acne increases for the first year or so, etc.) or estrogen-driven puberty.
Here are the various forms of administering hormones:
Top Surgery: a double mastectomy for trans masculine folks, and a breast augmentation for trans feminine folks.
Here are the several types of top surgery for trans masculine folks:
A lot of surgeons have slightly different techniques but these are the two main ones. There is also the T-anchor surgery which includes a vertical scar extending down an inch or so under each nipple. This preserve nipple sensation when paired with the Double Incision technique. On it's own, as a similar surgery to Peri-Areolar, it can aid in getting rid of excess skin and fat that gathers under the nipples.
Middle Surgery: the removal of one's reproductive organs - uterus, ovaries, fallopian tubes. Many people undergo this because it is a sort of spiritual release of one's original assigned sex, but it also can chemically aid in one's medical transition because ovaries produce estrogen which competes with the prescribed testosterone. Some people experience better results from the testosterone once their reproductive organs are removed. I have not chosen to undergo this as I have not had any problems with my testosterone or uterus thus far and feel no need to rid my body of these organs - but that's just me!
Bottom Surgery. The reconstruction of one's genitalia. Sometimes called "gender reassignment surgery" or "genital reassignment surgery" but many folks find these terms antiquated and/or inaccurate as reconstruction of genitalia does not constitute a "reassignment" of gender identity, but rather an affirmation of it. That is, I am a man regardless of whether or not I get bottom surgery. If I do choose to undergo it, this will be an act of affirmation, not an act of reassignment. For trans masculine folks, bottom surgery can include a vaginectomy (removal and sealing of the vaginal cavity), addition of a penis, and testicular implants.
Here are the two main types of bottom surgery for trans masculine folks:
Trans feminine folks can under go a variety of bottom surgeries as well, including vaginoplasty, orchiectomy, etc.
Facial Feminiziation Surgery (FFS): surgery for trans feminine folks to reconstruct parts of the face to be more gender affirmining.
Note: Do not ask trans people about their surgeries. This is invasive and unnecessary. See this post for more information. Rememeber that "The Surgery" doesn't exist. There are at least 14 different gender affirming surgeries people can get!
LASTLY: Again, everyone's transition is different and it is up to every individual to decide what their transition means to them. Medically transitioning isn't something everyone chooses to do -- in this world, it is also often an immense privilege, either financially or sociopolitically, or both. Those that do choose to medically transition don't all choose to undergo every surgery. A trans* identified person who hasn't gone through any or all of the surgeries or isn't on hormones is not any less of the gender they identify with. Surgery and hormones are a personal choice and do not define one's gender.