Please note that the above examples of possible gender identities are just examples, and are not meant to be all-inclusive.
Gender identity is your sense of who you are as a person - whether you are a man, a woman, a mix of the two, neither, or something separate from that binary entirely. For most people, this is relatively fixed; for some, it is somewhat fluid. There is considerable evidence suggesting that this sense of identity is rooted in the brain, and develops during gestation. The words "male" and "female" are frequently used to describe individuals' genders, though they are also frequently used to refer to biological sex.

Please note that the above examples of possible gender expressions are just examples, and are not meant to be all-inclusive.
Gender expression is about how you present yourself. It includes things like clothing, mannerisms, speech, and other gendered patterns of behavior. It is more or less the performative aspect of gender - how you "do" gender. It is not tied to gender identity: a male-identified person might very well prefer feminine gender expression.

Please note that the above examples of possible orientations are just examples, and are not meant to be all-inclusive.
Orientation has two main components: romantic or emotional orientation, and sexual orientation - each describing the type of person (or, in the case of sexual orientation, the type of parts) you are attracted to. These two characteristics are not always in alignment: for example, as above, it is possible for an asexual person to be heteroromantic, homoromantic, biromantic; or a person could be bisexual but heteroromantic, or panromantic but homosexual, etc. etc.

Biological sex is a complex issue. Here are some various approaches to defining it:

Some people consider that the question of whether a person's sex is male or female starts and ends with their chromosomes - a person with XY sex chromosomes is biologically male, a person with XX sex chromosomes is biologically female. However, this leaves out individuals with, for example, XXY sex chromosomes, and would contend that women with complete androgen insensitivity syndrome are biologically male and men with XX male (or de la Chapelle) syndrome are biologically female, as well as being problematic for a variety of other intersex conditions. Moreover, reducing sex to simply a matter of chromosomes renders the term fairly meaningless, as most people do not know which sex chromosomes they actually have. Some people instead look at genitalia as the be-all and end-all - this entails that any individual whose genitalia are for example destroyed in an accident are then neither biologically male nor biologically female. Some people place gamete production as the central defining factor, stating that in humans as with other organisms an individual that produces many small, mobile gametes (i.e. sperm) is male, whereas an individual that produces fewer large, relatively immobile gametes (i.e. eggs) is female. This approach implies that individuals who are born sterile are neither male nor female. Some people (including the creator of this visual aid) consider that the best way to look at an individual's biological sex is to consider their entire biology, and all of their sexually dimorphic traits, including any of the above as well as their secondary sex characteristics. By this model, the words "male" and "female" (as in "biologically male" and "biologically female") refer to ranges on either end of the continuum, rather than to discrete points; and it becomes meaningful to describe someone as "more female" or "more male", rather than simply either male, female, or neither, depending on whether they meet all of criteria A, or whether they meet all of criteria B, or whether they fail to meet at least one criterion from each category.