What it means to be a Jedi differs depending on whether you ask me as a part of the audience of Star Wars, a player of Star Wars: The Old Republic, or the individual characters I play.
Auset Midnight, my main character, twi'lek Infiltration Shadow it means the following:
She is a good spirit guarding the galaxy, showing up where she needs to be and help aid those in need, help sentient life flourish in all shapes throughout the galaxy and resolve conflicts. She is sworn to the Republic, but sees her foremost duty not towards any particular political entity, but to pluralistic galactic society of different peoples, only softly ruled in the name of all the planets and cultures (but not necessarily individuals) it comprises. She sees her role in ensuring those different cultures can develop coexistingly to the benefit of all. She helps settling conflicts between those, assist those in need with her powers, while respecting local authority.
As a member of a highly disciplined martial order, she holds a monopole of violence in times of peace, thus ensuring that it doesn't come to violence when she mediates between cultural differences and conflicting interests. However, due to her special abilities as shadow her speciality is infiltration, intelligence, and at times even assassination, of course, but that does not affect the way what it means to be a Jedi to her.
Swordsmanship and chivalry are important to her and sees it as privilege of the Jedi order. Perfecting her lightsaber skills is a way to perfect herself, it is not only a means to overcome her enemies, it's sport, art and a means to express herself in private situations.
Dealing with emotions is an important aspect of being a Jedi to her. To be truly at peace she must not try to not have emotions, but to understand them, be aware of them, even embrace them but not surrender to them. It's a delicate balance of accepting who she truly is, including all the flaws of her personality, all her desires and yet being considerate and calm when she lets them come to be. It takes a lot of reflection, counsel of other trusted Jedi and some private space to express herself for her. When she meditates it's often a lightsaber workout or gymnastic exercises, for her physical self is very important to her and she can't be one with herself without being one with her body, flowing with the Force to find peace.
To her it's not so that the Jedi are right and the Sith are wrong, it's merely a different approach, both conclusive and meaningful in their own way. She just chooses to guardian of a radiant, colorful galaxy, rather than being a radiant light herself, and overcoming the dark side within herself means making that choice anew with every breath she takes.
She trusts her instincts rather than denying them, allowing herself to fulfill the listen to the guidance of the force (though she would not go as far as to speak of a "Will of the Force" as Qui-Gon does) trying to always let it in and be aware of the Force around, dancing with it rather than taking control of it or watching it in suspicion.
What she hopes to achieve is difficult to answer - her goals are ultimately self perfection (physical and mental) and improved understanding and sharing that with the Jedi Order. And never become to proud to wonder, try to see the beauty of the galaxy every day anew, even in the darkest times. It's not something that can finish at some point though, it's an ongoing process.
Quotes from Star Wars with which she can identify herself best:
You will find that many truths we cling to depend greatly on our own point of view.
Feel, don't think.
Master Yoda says I should be mindful of the future. - But not at the expense of the moment.
Wow this is excellent!