Native deep cloning

It's called "structured cloning", works experimentally in Node 11 and later, and hopefully will land in browsers. See this answer for more details.

Fast cloning with data loss - JSON.parse/stringify

If you do not use Dates, functions, undefined, Infinity, RegExps, Maps, Sets, Blobs, FileLists, ImageDatas, sparse Arrays, Typed Arrays or other complex types within your object, a very simple one liner to deep clone an object is:

JSON.parse(JSON.stringify(object))

const a = {
  string: 'string',
  number: 123,
  bool: false,
  nul: null,
  date: new Date(),  // stringified
  undef: undefined,  // lost
  inf: Infinity,  // forced to 'null'
  re: /.*/,  // lost
}
console.log(a);
console.log(typeof a.date);  // Date object
const clone = JSON.parse(JSON.stringify(a));
console.log(clone);
console.log(typeof clone.date);  // result of .toISOString()

See Corban's answer for benchmarks.

Reliable cloning using a library

Since cloning objects is not trivial (complex types, circular references, function etc.), most major libraries provide function to clone objects. Don't reinvent the wheel - if you're already using a library, check if it has an object cloning function. For example,

• lodash - cloneDeep; can be imported separately via the lodash.clonedeep module and is probably your best choice if you're not already using a library that provides a deep cloning function
• AngularJS - angular.copy
• jQuery - jQuery.extend(true, { }, oldObject); .clone() only clones DOM elements
• just library - just-clone; Part of a library of zero-dependency npm modules that do just do one thing. Guilt-free utilities for every occasion.

ES6 (shallow copy)

For completeness, note that ES6 offers two shallow copy mechanisms: Object.assign() and the spread syntax. which copies values of all enumerable own properties from one object to another. For example:

var A1 = {a: "2"};
var A2 = Object.assign({}, A1);
var A3 = {...A1};  // Spread Syntax

To remove a property from an object (mutating the object), you can do it like this:

delete myObject.regex;
// or,
delete myObject['regex'];
// or,
var prop = "regex";
delete myObject[prop];

Demo

var myObject = {
    "ircEvent": "PRIVMSG",
    "method": "newURI",
    "regex": "^http://.*"
};
delete myObject.regex;

console.log(myObject);

For anyone interested in reading more about it, Stack Overflow user kangax has written an incredibly in-depth blog post about the delete statement on their blog, Understanding delete. It is highly recommended.

If you'd like a new object with all the keys of the original except some, you could use the destructuring.

Demo

let myObject = {
  "ircEvent": "PRIVMSG",
  "method": "newURI",
  "regex": "^http://.*"
};

const {regex, ...newObj} = myObject;

console.log(newObj);   // has no 'regex' key
console.log(myObject); // remains unchanged