The general template for an Anglophone name is:
Title Given-name(s) Surname
"White" is his surname. Surnames are family names and are generally inherited. Traditionally this has been focussed on the male line - a woman would take the man's surname when they marry, and their children would usually take the man's surname too. (This is not entirely universal though. Sometimes a woman called Jones marrying a Mr Smith may, instead of changing her name to Smith, adopt a combined surname Smith-Jones, and their children would use that. Or she may simply continue to use Jones.)
Many English surnames are occupational (Thatcher, Baker, Cooper), some are patronymic (Johnson, Wilson), some are descriptive (Short), some come from geographical placenames or descriptors (Wood, Meadows).
Then we have given names, also called forenames or Christian names. (Many of these names are biblical in origin, but we often use the term Christian name even for non-biblical names of atheist/non-Christian people, it's just a linguistic habit.) Some people have only one given name (John Smith), others have many (John Edward William David Smith). Walter White appears to have two, Walter Hartwell, which is common. In this instance we may refer to 'Hartwell' as his middle name. It is quite common, although by no means universal, for one or more of child's given names to be that of a parent, grandparent etc.
Usually, people will only use one of those names as their day-to-day name. And usually, it will be the first - in this case Walter. This is not necessarily the case though. It is relatively common for (e.g.) John David Smith to go aroundthrough life consistently presenting himself as 'David', even having "David Smith" on a workplace ID card for example. For example sports commentator Murray Walker is actually named Graeme Murray Walker.
Additionally, we also commonly vary the given names into nicknames and diminuitives. For example David Smith's friends and family may call him Dave. Such names are potentially a marker of closeness and familiarity. He may prefer to be called David at work, or by strangers, and consider the use of 'Dave' to be something that is exclusively reserved for his friends. On the other hand, he may also simply prefer 'Dave' in all situations.
Then we have titles. Historically these could denote man (Mister/Master) versus woman (Mrs/Miss), married (Mister/Mrs) versus unmarried (Master/Miss), and social rank or status (Sir, Lady, etc). In modern times some or even all of those aspects are somewhat controversial. "Mr" is now used as a basically universal term for men, without any implications about their marital or social status, while some favour "Ms" as a similarly 'neutral' female counterpart.
So finally reaching the actual question, why do people call him different things.
In formal and business situations "Title Surname" (Mr White, Mrs Smith, etc) is a standard format of respectful address. In everyday informal situations however we would expect to use simply their favoured forename (Walter), or a nickname variation of it ("Walt").
The full name ("Walter Hartwell White") is probably rarely used in spoken conversation, and mostly appears when filling in forms and paperwork. Stereotypically, the sound of someone (particularly a parent, teacher or authority figure) spelling out your full name has connotations of you being in big trouble!
(Of course please note the customs and traditions around people's names are extremely diverse, so nothing I say here is a solid rule.)
