SharePoint 2010 file upload via REST api

You can use Copy Web Service method to upload document in SharePoint 2010 But it is not a method of REST API.

Below is the article from where you will get all the details.

Uploading Content Into SharePoint (Let me count the ways)

Below is the code WORKING in SharePoint 2013 to upload a file in the same domain by using the REST API and jQuery

'use strict';

jQuery(document).ready(function () {

    // Check for FileReader API (HTML5) support.
    if (!window.FileReader) {
        alert('This browser does not support the FileReader API.');
    }
});

// Upload the file.
// You can upload files up to 2 GB with the REST API.
function uploadFile() {

    // Define the folder path for this example.
    var serverRelativeUrlToFolder = '/shared documents';

    // Get test values from the file input and text input page controls.
    var fileInput = jQuery('#getFile');
    var newName = jQuery('#displayName').val();

    // Get the server URL.
    var serverUrl = _spPageContextInfo.webAbsoluteUrl;

    // Initiate method calls using jQuery promises.
    // Get the local file as an array buffer.
    var getFile = getFileBuffer();
    getFile.done(function (arrayBuffer) {

        // Add the file to the SharePoint folder.
        var addFile = addFileToFolder(arrayBuffer);
        addFile.done(function (file, status, xhr) {

            // Get the list item that corresponds to the uploaded file.
            var getItem = getListItem(file.d.ListItemAllFields.__deferred.uri);
            getItem.done(function (listItem, status, xhr) {

                // Change the display name and title of the list item.
                var changeItem = updateListItem(listItem.d.__metadata);
                changeItem.done(function (data, status, xhr) {
                    alert('file uploaded and updated');
                });
                changeItem.fail(onError);
            });
            getItem.fail(onError);
        });
        addFile.fail(onError);
    });
    getFile.fail(onError);

    // Get the local file as an array buffer.
    function getFileBuffer() {
        var deferred = jQuery.Deferred();
        var reader = new FileReader();
        reader.onloadend = function (e) {
            deferred.resolve(e.target.result);
        }
        reader.onerror = function (e) {
            deferred.reject(e.target.error);
        }
        reader.readAsArrayBuffer(fileInput[0].files[0]);
        return deferred.promise();
    }

    // Add the file to the file collection in the Shared Documents folder.
    function addFileToFolder(arrayBuffer) {

        // Get the file name from the file input control on the page.
        var parts = fileInput[0].value.split('\\');
        var fileName = parts[parts.length - 1];

        // Construct the endpoint.
        var fileCollectionEndpoint = String.format(
                "{0}/_api/web/getfolderbyserverrelativeurl('{1}')/files" +
                "/add(overwrite=true, url='{2}')",
                serverUrl, serverRelativeUrlToFolder, fileName);

        // Send the request and return the response.
        // This call returns the SharePoint file.
        return jQuery.ajax({
            url: fileCollectionEndpoint,
            type: "POST",
            data: arrayBuffer,
            processData: false,
            headers: {
                "accept": "application/json;odata=verbose",
                "X-RequestDigest": jQuery("#__REQUESTDIGEST").val(),
                "content-length": arrayBuffer.byteLength
            }
        });
    }

    // Get the list item that corresponds to the file by calling the file's ListItemAllFields property.
    function getListItem(fileListItemUri) {

        // Send the request and return the response.
        return jQuery.ajax({
            url: fileListItemUri,
            type: "GET",
            headers: { "accept": "application/json;odata=verbose" }
        });
    }

    // Change the display name and title of the list item.
    function updateListItem(itemMetadata) {

        // Define the list item changes. Use the FileLeafRef property to change the display name. 
        // For simplicity, also use the name as the title. 
        // The example gets the list item type from the item's metadata, but you can also get it from the
        // ListItemEntityTypeFullName property of the list.
        var body = String.format("{{'__metadata':{{'type':'{0}'}},'FileLeafRef':'{1}','Title':'{2}'}}",
            itemMetadata.type, newName, newName);

        // Send the request and return the promise.
        // This call does not return response content from the server.
        return jQuery.ajax({
            url: itemMetadata.uri,
            type: "POST",
            data: body,
            headers: {
                "X-RequestDigest": jQuery("#__REQUESTDIGEST").val(),
                "content-type": "application/json;odata=verbose",
                "content-length": body.length,
                "IF-MATCH": itemMetadata.etag,
                "X-HTTP-Method": "MERGE"
            }
        });
    }
}

// Display error messages. 
function onError(error) {
    alert(error.responseText);
}

Check this article for more details.

Upload a file by using the REST API and jQuery