Depending on your situation, there are a few different approaches. I can think of four different ways to conditionally require a field.

Dependencies

The dependentSchemas keyword is a conditional way to apply a schema. Foreach property in dependentSchemas, if the property is present in the JSON being validated, then the schema associated with that key must also be valid. If the "foo" property is present, then the "bar" property is required

{
  "type": "object",
  "properties": {
    "foo": { "type": "string" },
    "bar": { "type": "string" }
  },
  "dependentSchemas": {
    "foo": { "required": ["bar"] }
  }
}

If all the dependent schema needs is the required keyword, you can use the dependentRequired keyword as a shorthand. The following has the same effect as the previous example.

{
  "type": "object",
  "properties": {
    "foo": { "type": "string" },
    "bar": { "type": "string" }
  },
  "dependentRequired": {
    "foo": ["bar"]
  }
}

NOTE: In draft-07 and below these were one keyword called dependencies. If the value is a schema it behaved like dependentSchemas. If the value is an array, it behaved like dependentRequired.

Implication

If your condition depends on the value of a field, you can use a boolean logic concept called implication. "A implies B" effectively means, if A is true then B must also be true. Implication can also be expressed as "!A or B". Either the "foo" property does not equal "bar", or the "bar" property is required. Or, in other words: If the "foo" property equals "bar", Then the "bar" property is required

{
  "type": "object",
  "properties": {
    "foo": { "type": "string" },
    "bar": { "type": "string" }
  },
  "anyOf": [
    {
      "not": {
        "properties": {
          "foo": { "const": "bar" }
        },
        "required": ["foo"]
      }
    },
    { "required": ["bar"] }
  ]
}

If "foo" is not equal to "bar", #/anyOf/0 matches and validation succeeds. If "foo" equals "bar", #/anyOf/0 fails and #/anyOf/1 must be valid for the anyOf validation to be successful.

NOTE: The if/then keywords have the same behavior, but are easier to read and maintain. It's recommended to only use this approach if you are using an older version of JSON Schema that doesn't support if/then.

Enum

If your conditional is based on an enum, it's a little more straight forward. "foo" can be "bar" or "baz". If "foo" equals "bar", then "bar" is required. If "foo" equals "baz", then "baz" is required.

{
  "type": "object",
  "properties": {
    "foo": { "enum": ["bar", "baz"] },
    "bar": { "type": "string" },
    "baz": { "type": "string" }
  },
  "anyOf": [
    {
      "properties": {
        "foo": { "const": "bar" }
      },
      "required": ["bar"]
    },
    {
      "properties": {
        "foo": { "const": "baz" }
      },
      "required": ["baz"]
    }
  ]
}

NOTE: This approach is not recommended because it can produce confusing error messages. The if/then keywords are generally a better approach.

If-Then-Else

The if, then and else keywords are shorthand for the implication pattern described above. These keywords were added in draft-07. If the "foo" property equals "bar", Then the "bar" property is required

{
  "type": "object",
  "properties": {
    "foo": { "type": "string" },
    "bar": { "type": "string" }
  },
  "if": {
    "properties": {
      "foo": { "const": "bar" }
    },
    "required": ["foo"]
  },
  "then": { "required": ["bar"] }
}

EDIT 12/23/2017: Implication section updated and If-Then-Else section added.

EDIT 06/04/2018: Bugfix for If-Then-Else and update singleton enums to use const.

EDIT 07/06/2022: Update Dependencies section to use the new dependentSchemas/dependentRequired keywords instead of dependencies.

The if keyword means that, if the result of the value schema passes validation, apply the then schema, otherwise apply the else schema.

Your schema didn't work because you needed to require "foo" in your if schema, otherwise an empty JSON instance would pass validation of the if schema, and therefore apply the then schema, which requires "bar".

Second, you want "propertyNames":false, which prevents having any keys in the schema, unlike if you were to set "else": false which would make anything always fail validation.

{
  "type": "object",
  "properties": {
    "foo": {
      "type": "string"
    },
    "bar": {
      "type": "string"
    }
  },
  "if": {
    "properties": {
      "foo": {
        "enum": [
          "bar"
        ]
      }
    },
    "required": [
      "foo"
    ]
  },
  "then": {
    "required": [
      "bar"
    ]
  },
  "else": false
}