Trans- is used both as a prefix to "gender" or "sexual", and on its own as an adjective.

"Transsexual" or "transexual" (the number of 's's in the middle varies depending on whom you ask) describes modifying or having modified the sex attributes of one's body. 

"Transgender" is used with various meanings.  Sometimes it is used as an "umbrella" term for anyone who is not cis or is gender variant in any way.  Sometimes it is used to mean the same thing as "transexual".  

"Trans", as an adjective by itself, does not specify whether it refers to "transexual" or "transgender". 

When using any of these words, it is important to remember that they are adjectives, rather than nouns.  It is much more respectful to talk about "transexual people" than "transexuals", for example. Similarly, "trans" can be used as an adjective to modify "person" or "woman" or "man", as in "trans woman" or "trans man".  Avoiding the prefix form of "trans-" here avoids gendering people as "transwoman" or "transman" and thus implicitly denying them the genders of "woman" or "man".