Since you're here, I'll assume you've got a basic grasp of what therianthropy is. If you want in-depth details, go to Google
and type in "therianthropy". I'm sure you'll find a good many essays. For now, we'll define "therianthropy" as "having a connection, be it spiritual or mental, with a certain species of animal. Many (but not all) experience certain types of 'shifts': mental, phantom, dream and astral" (I omit "physical" as I do not believe it's possible) and go from there.
Lots of "therianthropes", "weres" or "lycanthropes" will tell you at one point or another that their spirit is part animal and that is their connection to that animal. While I don't believe my spirit is anything but human, I do associate myself with cougars. I believe it's more of a "mental" connection, meaning that I do have the mindset and some habits of this gorgeous cat. This isn't to say that I go around attacking people and/or animals thinking that I am a cougar, rather that the cougar is mentally a part of me.
The "character" that represents this part of me is in the form of an anthromorphic cougar that I have named "Rynoah". In some ways, she is like a second personality and in others, she is just me in a different form. As mentioned above, "therianthropes" experience what is commonly referred to as "shifts" (unless you're a "contherianthrope", in which case you are in a constant state of mental shift. Enter "contherianthrope" into any search engine for details). I am no different, my main shifts usually "mental" and the occasional "dream" and the very, very rare "phantom". When Rynoah most usually is a second personality is when I "mentally shift" (an experience where a "therianthrope" reverts to a state of their animal's mentality and/or behavior. These are very rarely overly violent and those that claim they performed a deed of malice without provocation because of their "wereside" are those that I view with skepticism). During "m-shifts" (as they're called short-hand), I am usually quite quiet and collected. I test the air for scents and sounds quite more than I depend on eyesight (unless in darkness, when I depend on all three) and tend to make more "purring" or growling noises than usual. It's hard to explain what I experience during mental shifts, for they are varied, so I'm afraid I'll have to leave that to your imagination. Though it's not entirely possible to predict when a mental shift comes on, usually situations where I am under stress or I am alone trigger the "wereside" to manifest itself. It's highly unusual that you'll actually see me mentally shift unless you observe me for a long time in a public place when I am sitting off by myself seemingly doing nothing (or you provoke me into a fight, which has rarely happened, so I can't guarantee you'll get the "wereside" fighting you.)
I have had very few "dream shifts", so I can't discuss much about them. Dream shifts are when a "therianthrope" dreams they are their animal in any shape or form. Usually in "dream shifts", I am actually a large black cat with saber-teeth (something I created called a "sabrok"), but I've had the rare "cougar" d-shift. These usually involve battle and I am usually Rynoah in an anthromorphic (or "crinos" if you're a White Wolf fan) form leading an army into war. These are short and--as I said--very rare, usually ending just before I cut into my first foe. The warriors I'm up against are usually anthromorphic wolves, so I don't really read much into these dreams. During these shifts, Rynoah and I are usually one, but my "human side" is the lesser side of the two of us. It's more like I'm along for the ride and not taking an active part in the control of our body.
Phantom shifts are experiences where a "therianthrope" can feel a limb or limbs that belong to their "wereside", quite like tales of amputees that can still feel their amputated limb long after it's gone. I've only had two of these and they were very faint and lasted but a moment. The first instance, I had lay in bed on the verge of dozing off when I felt what I can only explain as a tail attached to my rump slowly twitching its tip. During the most recent, I had the feeling of ears atop my head that I could swivel and flick with a thought. It was a great experience and I wish I could make it happen again, but I never "force" my wereside to manifest herself no matter what form it may be.
One of many false beliefs about "therianthropy" is that it is a cult and should be viewed with a hint of caution. These are gross misinterpretations. Therianthropy is a personal spirituality. The only reason the "community" as a whole was created was because a few "therianthropes" got together and decided it'd be cool to hang out because of their similar beliefs. I believe you'd find more cult-like activities in a group of gamers talking about a favorite game than you would in the "were'" community.
In short, the cougar is part of me.