Standard ML (SML) is a general-purpose programming language designed for use in large-scale projects. It was initially developed in the late 1980s at the University of Edinburgh, and since then, it has gained popularity as an academic language for teaching functional programming concepts. SML has a strong static type system that helps catch errors at compile-time, making it an ideal language for writing reliable and maintainable software. In this beginner's guide, we'll take a look at SML's syntax, show you how to write a basic program, and explore some of the most common applications of the language.

Syntax:
SML has a concise and uniform syntax that makes it easy to read and write. The language is made up of expressions, which are built from values, variables, and functions. Here is a simple example of an SML expression:

```
fun square(x : int) : int = x * x;
```

This expression defines a function called `square` that takes an integer parameter `x` and returns the square of `x`. The `: int` after the parameter and return types specifies their types as integer. The `fun` keyword is used to define a function, and the `=` separates the function signature from its implementation.

Variables in SML are immutable, meaning they cannot be changed once they are assigned a value. Here is an example of how to declare a variable in SML:

```
val name : string = "John Doe";
```

This declares a variable called `name` with a type of `string` and assigns it the value "John Doe". The `val` keyword is used to declare a value.

Writing a Basic Program:
Let's take a look at a simple SML program that reads in an integer value and prints out its square:

```
fun main() =
    let
        val x = Int.fromString(TextIO.inputLine(TextIO.stdIn))
    in
        print("The square of " ^ Int.toString(x) ^ " is " ^ Int.toString(x*x) ^ "\n")
    end;
```

This program defines a function called `main` that reads in an integer from standard input, calculates its square, and prints out the result. The `^` operator is used to concatenate strings. 

To run this program, you'll need an SML compiler. One popular SML compiler is the Standard ML of New Jersey (SML/NJ) compiler, which can be downloaded from the official website.

Applications:
SML is commonly used for research in programming language theory and functional programming. It is also used in the development of compilers, theorem provers, and other software tools. Here are some examples of notable applications of SML:

1. MLton: An optimizing whole-program compiler for SML that produces fast and efficient code.

2. HOL4: A higher-order logic theorem prover developed in SML that is used for formal verification.

3. SML/NJ Library: A collection of libraries and tools developed in SML for various tasks such as regular expressions, parsing, and I/O operations.

4. ddcML: A tool for generating high-performance C code from SML programs.

Conclusion:
Standard ML is a powerful programming language that is particularly well-suited for developing large-scale, reliable software. Its concise syntax and strong type system make it a joy to write and easy to read. In this beginner's guide, we've covered the basics of SML programming, including its syntax, how to write a basic program, and some of its most common applications. With this knowledge, you should be well on your way to mastering SML and developing your own high-quality software.