What is the relationship among these patterns?

In your application, you will use many design pattern. It rarely happen that you will use only one or none design pattern, until you are  you are making a+b program.
Generally, to build a system, you may need many patterns to fit together. Different designer may use different patterns to solve the same problem. Usually:

• Some patterns naturally fit together
• One pattern may lead to another
• Some patterns are similar and alternative
• Patterns are discoverable and documentable.
• Patterns are not methods or framework
• Patterns give you hint to solve a problem effectively

How many design patterns?

There are many design patterns. A site says at least 250 existing patterns are used in OO world, including Spaghetti which refers to poor coding habits. The 23 design patterns by GOF are well known, and more are to be discovered on the way.
Note that the design patterns are not idioms or algorithms or components.

Spaghetti  Code: Spaghetti code is a pejorative term for source code that has a complex and tangled control structure, especially one using many GOTOs, exceptions, threads, or other "unstructured" branching constructs. It is named such because program flow tends to look like a bowl of spaghetti, i.e. twisted and tangled. Spaghetti code can be caused by several factors, including inexperienced programmers and a complex program which has been continuously modified over a long life cycle. Structured programming greatly decreased the incidence of spaghetti code.

Need Of Design Pattern(Do I have to use the design pattern?)

If you want to be a professional Java developer, you should know at least some popular solutions to coding problems. Such solutions have been proved efficient and effective by the experienced developers. These solutions are described as so-called design patterns.

Many programmers with many years experience don't know design patterns, but as an Object-Oriented programmer, you have to know them well, especially for new Java programmers. Actually, when you solved a coding problem, you have used a design pattern. You may not use a popular name to describe it or may not choose an effective way to better intellectually control over what you built. Learning how the experienced developers to solve the coding problems and trying to use them in your project are a best way to earn your experience and certification.

Design Pattern have two main usages:

• Common language for developers: They provide developer a common language for certain problems. For example if a developer tells another developer that he is using a Singleton, the another developer (should) know exactly what this means.
  
• Capture best practices: Design patterns capture solutions which have been applied to certain problems. By learning these patterns and the problem they are trying to solve a unexperienced developer can learn a lot about software design.
  
  
  Design pattern are based on the base principles of object orientated design.
  
  • Program to an interface not an implementation
    
  • Favor object composition over inheritance