This page in the documentation covers Triggered Sends: Triggered Email Scenario Guide For Developers
In it's simplest form, you can do a form POST to an SFMC endpoint with an XML SOAP envelope like this:
<s:Envelope xmlns:s="http://www.w3.org/2003/05/soap-envelope" xmlns:a="http://schemas.xmlsoap.org/ws/2004/08/addressing" xmlns:u="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-utility-1.0.xsd">
<s:Header>
<o:Security s:mustUnderstand="1" xmlns:o="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-secext-1.0.xsd">
<o:UsernameToken u:Id="uuid-f4119202-0c6f-4383-9872-4745b39f24f6-1">
<o:Username>XXXXX</o:Username>
<o:Password>XXXXX</o:Password>
</o:UsernameToken>
</o:Security>
</s:Header>
<s:Body xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema">
<CreateRequest xmlns="http://exacttarget.com/wsdl/partnerAPI">
<Options>
<RequestType>Asynchronous</RequestType>
<QueuePriority>High</QueuePriority>
</Options>
<Objects xsi:type="TriggeredSend">
<PartnerKey xsi:nil="true"/>
<ObjectID xsi:nil="true"/>
<TriggeredSendDefinition>
<PartnerKey xsi:nil="true"/>
<ObjectID xsi:nil="true"/>
<CustomerKey>ETWSEX_TSD</CustomerKey>
</TriggeredSendDefinition>
<Subscribers>
<PartnerKey xsi:nil="true"/>
<ObjectID xsi:nil="true"/>
<EmailAddress>[email protected]</EmailAddress>
<SubscriberKey>[email protected]</SubscriberKey>
<Attributes>
<Name>FromName</Name>
<Value>From John Doe</Value>
</Attributes>
<Attributes>
<Name>HTML__Content</Name>
<Value><b>Sent Asynchronously!</b></Value>
</Attributes>
</Subscribers>
</Objects>
</CreateRequest>
</s:Body>
</s:Envelope>
If you'd like to utilize the Java SDK that SFMC provides, you can also do it that way, but it's more involved.
If you're familiar with REST APIs, then you can also trigger emails with SFMC's REST API.