When any character is introduced in general, it is important to note his/her significance in the story itself. Is the character simply part of the setting (and thus not meeting the criteria of even a secondary character)? Is the character designed to support the story in some way, whether that be other characters, the conflict, central themes, et cetera? Is the character designed for a more important and individual reason; or, in other words, does the character exist for its own sake? Having a clear understanding of how you want your character to function in your story is integral to understanding how to introduce him/her. Remember, introductions are important for first impressions and are generally important for helping establish a character in any given work.
Typically, for me, when I introduce any primary or secondary character, I try not to make the introduction seem too random (unless I am trying to emphasize a theme like the fact that we sometimes meet people spontaneously). I try to have the writing lead to the character's necessity. This can be done in a multiple of ways. You can have the character be known to exist while shielding any details about him/her until the story necessitates that the character be introduced either directly or indirectly. You can substantiate a character by have him/her be an anthropomorphic representation of a plot thread or a conflict. An example of this can be the situation in which the protagonist enters upon some man-versus-man, say a castle siege. When the protagonist goes to analyze the situation, he/she will naturally end up meeting those directly involved in the conflict itself. In general, you want to write the build up for your new character to seem natural. You typically don't want the character to seem sudden, unexpected, or flow-breaking with your narrative. The exception obviously being unless you explicitly want your character to seemingly pop out of nowhere.
I haven't answered your question though, you want to know how to introduce your character. All I've done is given examples of giving pretext. One important aspect I always want to make sure I express is the character's speech, tone, and attitude. If I introduce the character, I want to differentiate his/her dialogue from the other characters fairly quickly. This may sound a bit too obvious, but characterize your characters. Subtly have the key characteristics of your character observable within his/her's introductory dialogue. Writers don't want their readers to become confused in conversations and they don't want to have to end each line of dialogue with, "X said." This sort of characterization is what I tend to do. I show the differences and uniqueness of new characters while slowly elaborating on the similarities (which thus give a justification for the protagonist to have any relation with the new character in the first place). Think about why your character is necessary, what does he/she bring to the story, and the different insight/perspective that the character will bring.
Also, if I may point out, take advantage of differences not only in personality, but in all realms of being. Differences in gender, cultural background, race, religion, upbringing, class, et cetera. These are great for giving us context into this character. If you share with us that this new character is the son of a governor from Tennessee, your reader can make some generalizations pretty quickly. Generalizations are not necessarily bad for the reader to make, they can be important for initially filling the void of knowledge that your protagonist, as well as your audience, has.
Specifically towards your necessity for casualness, flesh your desire out. If you want a casual introduction, then you need to think about what that entails. What is a casual introduction? In what contexts are casual introductions made? Is a casual introduction formal or informal, possibly both? Do you want to emphasize the casual nature of their first meeting or do you want the casualness to be only implicit? Understanding the criteria of exactly what you want is very important to actually getting it.
And, finally, once you have written your introduction, rewrite it. Writers are human beings, we make mistakes, we have skips in train of thought, we make chronological errors, we end up writing what we wanted to avoid. The act of rewriting is necessary because you will almost never translate what you had imagined perfectly onto paper. That is the core nature of thoughts. They are vague, misleading, and not properly reasoned. They are only an idea and not the actual implementation. So once you write your introduction, rewrite it. This will help ensure good prose, brevity, appropriate detail, and, more specifically towards your question, a succinct expression of your new character.