Arthur; the point man

A point man is a person who "has a crucial, often hazardous role in the forefront of an enterprise" or a "soldier who is assigned to a position some distance ahead of a patrol as a lookout." Arthur is the point man of the team (as well as always being Cobb's point man and second in most extrations) because he always does the research needed into their mark's, or subject's, past and he (along with Yusuf) also does a lot of the important coordination work in the film. In the movie, Arthur is characterized as very practical, no nonsense, and strait-laced, often dressed in suits and clean-cut clothing, with his hair slicked back neatly.
Eames; the forger

A forger in the world which Inception takes place is a person who can imitate and impersonate someone else within a dream. This often requires the forger to observe his or her target for a period of time so as to learn that person's habits and mannerisms. Forgers are often connected with counterfeiting and theft. When Eames is first mentioned on screen, he is referred to as a good thief, but Cobb risks capture to bring Eames on board because he is a forger. When we meet Eames in Mombasa, he stands a stark contrast to Arthur's strait-laced demeanor. Eames is characterized as flippant and a charmer, often dressing in colorful clothing. He admits to being poor in spelling and math.

Arthur + Eames
The first really major moment between Arthur and Eames was the final scene in the second trailer, where even before one sees the film, their bickering can be felt. In the movie itself, the first time Eames is mentioned is by Cobb, Arthur dismisses the idea because "There's plenty of good thieves," and where Eames is at (Mombasa) is a dangerous place for them, but Cobb says, "We don't just need a thief. We need a forger." When Cobb finds Eames, after hearing Cobb mention Arthur, Eames comments with, "Arthur... You're still working with that stick in the mud?" Later Eames admits Arthur is the best at what he does, but lacks imagination, unlike himself.
Later on, when the team plans how they will perform the inception, Eames comes up with many creative ideas while Arthur is there to question the details and cast skepticism. They have many back and forth exchanges where we can see Eames likes to be shaky with details while Arthur prefers to plan out everything possible, leaving little to chance. They don't necessarily see eye to eye on many things, but they both share a passion for their line of work that enables them to work well together despite their personal differences.