URL Redirect

This record type is used when you wish to configure your domain in the following way: when the users type the name of your domain in the address bar, they are redirected to any specific web page, for example, http://maps.google.com/ or http://en.wikipedia.org/wiki/Main_Page
The domain is redirected to the indicated page and in URL bar the destination address is shown instead of yourdomain.com.
This is how the record should be configured in Namecheap account, All Host Records section:

URL Frame

This host record is used when you wish to redirect your domain name similarly to URL redirect, but you wish to have the destination link to be shown in browser instead of your domain.
Please see how it should be configured in All Host Records section:

URL Redirect (301)

This option should be used when you wish to permanently redirect your domain to some specific URL.
In this case search engines are notified of a permanent address change and the values of internal links are transferred to the new site. Please select URL Redirect (301) from the Record Type drop-down menu.

A (Address) record (IPv4)

This record should be created when you want to associate your domain or subdomain with IP address (32-bit one). Please note that you can only set up A record for the domain if it's using our default nameservers.
Please enter an IP address for both the WWW and the @ fields and choose A (Address) as the Record Type from the drop down menu:

AAAA record (IPv6)

Please use this record if you wish to associate your domain or subdomain with IP address (128-bit one). It is configured the same way A record is, but in Record Type menu AAAA (IPv6) one should be selected:

CNAME record

This record specifies that the domain name in question is an alias of another, canonical name.
CNAME record will point your domain or subdomain to the IP address of the destination hostname. So, if the IP of the destination hostname changes, you won't need to change your DNS records as the CNAME will have the same IP.
Please don't set up CNAME for naked domain (@ hostname), since it may affect the operation of the domain's MX records and, consequently, the email service. Thus, in most cases you will need to create CNAME record for WWW (or other subdomain) and URL redirect for @ that will point to http://yourdomain.com/

MX record

MX is used for forwarding e-mail to a mail server name i.e. mail.mailserver.com.
If you have a mail server at mail.somemailserver.com and you want to add this for your domain name, you need to add  MX record for @ hostname, and indicate mail.yourhost.com for mailserver host name. This would mean that any emails addressed to @yourdomain.com would be delivered to the server at mail.yourhost.com.
You also need to set priority for your MX record in MX Pref column.

MXE record

MXE (Mail Easy) is used for forwarding mail to an IP address.
Please also set priority for your MXE record in MX Pref column.

TXT record

TXT record is used to hold some text information. You can input any text for descriptive purposes. TXT records are often used to store SPF (Sender Policy Framework) records and to prevent people from receiving fake emails. Sometimes this records needs to be set up for verification purposes, for example, some Google services require it. It is reflected in the screen shot below:

SRV record

SRV records allow you to define the location, i.e. the hostname and port number, of servers for specified services.

NS record

This type of record allows you to delegate a subdomain of your domain to the nameserver, not associated with the domain itself. This is needed when your subdomain is hosted separately from the domain itself.