Bhakti is devotion. Feelings of respect, love and Shraddhaa towards Ved, Shaastra and Parameshwar, saints, and Guru is called Bhakti. It can be simply listening to discourse, having Darshan, touching Bhagavaan's feet, or staying in the company of Bhagavaan. In Tulasee's Maanas, Raam describes nine types of Bhakti (Navadhaa Bhakti).

[Aangiras, p 256-257] Devotion, especially to one's personal God. The devotee can consider his relationship with God in various forms - (1) that of a servant to his master, such as Hanumaan to Raam; (2) that of a child to his parents as that of Prahlaad to Naaraayan; (3) that of a parent to child as that of Yashodaa to Krishn; (4) that of a friend to a friend as that of Arjun to Krishn; (5) as that of a cow to her cowherd as that of Gopee of Vrindaa Van to Krishn; (6) even that of a wife to her husband as that of Rukminee or Meeraa to Krishn. Bhakti consists of love and trust. This human emotion is as old with human beings as the relationship between a man and a woman who love and trust each other.

Total Bhakti means no ego, no self identity. The self does everything for God. To get there one has to sacrifice the ego and selfishness.

Bheeshm Jee asked Maharshi Pulastya - "Who are called Brahmaa's Bhakt?" Pulastya Jee said - "Bhakti is of three types - Maanas, Vaachik and Kaayik (by mind, by speech and by body). Besides this, there are three types of Bhakti more - Laukik, Vaidik and Adhyaatmic. When Ved meanings are thought by Dhyaan, that is called Maanas Bhakti. When one does Mantra Jap, Ved reading, and doing Jap in forests, it is called Vaachik Bhakti. When one worships Bhagavaan by keeping fasts, doing Vrat, discipline, Chaandraayan Vrat etc, that is called Kaayik Bhakti. These types of Bhakti is told for Dwijaati people.
[Padm Puraan, 1/12]

Who worship Brahmaa Jee with cow's Ghee, milk, Yogurt, Dhoop, ornaments, fruits, Naivedya etc, that is called Laukik Bhakti. Who worships him by Ved Mantra Jap and teaching Sanhitaa etc, it is called Vaidik Bhakti. When Havan is done by offering Aahuti of Havishya pronouncing Ved Mantra, that is also called Vaidik Bhakti. Whatever Agnihotra is done on Poornimaa and Amaavasyaa, whatever Dakshinaa is given in Yagya, whatever Kheer is offered to Devtaa etc, that also come under Vaidik Bhakti. Whatever worship is done for Agni, Prithvi, Vaayu, Aakaash, Chandramaa etc, it all goes to Brahmaa. Brahmaa's Adhyaatmic Bhakti is of two types - Saankhyaj, and Yogaj. Pradhaan, etc Praakrit elements are 24 in number. They all are Jad (unconscious) and consumable. Their enjoyer Purush (here it is referred to man) is the 25th element. He is conscious. To know all these elements properly is known as Saankhyaj Bhakti. Learned people consider it Adhyaatmic Bhakti under Saankhya Shaastra. Who meditates upon Brahmaa Jee's figure by Praanaayaam and controls his Indriyaan, it is called Yogic Bhakti, or Paraa Bhakti. Whoever does this kind of Bhakti, he is called Brahmaa Jee's Bhakt.
[From Padm Puraan, 1-Srishti Khand, p 44-45]

Ambareesh said to Naarad Jee - "Now I want to know something about Bhakti that which man should do which type of Bhakti when, where and how?" Naarad Jee said - "Bhagavaan's Bhakti destroys all kinds of sins. Bhakti is of several types - 

(1) Maanasee Bhakti (by mind) - When Vishnu is pleased by Dhaaranaa, intellect and understanding the meaning of Ved, it is called Maanasee Bhakti.
(2) Vaachikee Bhakti (by speech) - When Bhagavaan is pleased by pronouncing Ved Mantra, Jap, and reading Aaranyak, it is called Vaachikee Bhakti.
(3) Kaayikee Bhakti (by body) - When Bhagavaan is pleased by Vrat (taking any vow), Upavaas (fast), following some discipline, controlling one's five Indriyaan, it is called Kaayikee Bhakti.
(4) Laukikee Bhakti (worldly) - Serving Bhagavaan's feet by Arghya, dance, playing musical instruments, singing, Jaagaran, and worshipping etc is called Laukikee Bhakti.
(5)  Vaidikee Bhakti (according to Ved) - Pleasing Bhagavaan by Jap of Ved Mantra, study of Sanhitaa, offering Aahuti of Havishya in Yagya, or doing Yagya is called Vaidikee Bhakti.
(6) Adhyaatmikee Bhakti (spiritual) - Pleasing Bhagavaan by doing Yagya on Amaavasyaa, Poornimaa, Vishuv Yog (Mesh and Tulaa - Aries and Libra, Sankraanti) is called Adhyaatmikee Bhakti.
[From Padm Puraan, 4/19, p 580]

Paarvatee Jee asked - "Hey Vishweshwar, What is the form of the best Bhakti?" Mahaadev Jee said - "Bhakti has been told of three types - Saatwik, Raajasik and Taamasik. Among them Saatwik is the best, Raajasik is medium and the Taamasik is the lowest kind of Bhakti. Who desire for Moksh, they should do Saatwik type of Bhakti of Shree Hari. Whatever Bhakti is done with pride, or to boast himself, or with jealousy, or to destroy somebody, that kind of Bhakti is called Taamasik. Who does Bhakti with worldly desires or to get wealth, it is called Raajasik Bhakti. Whatever Bhakti is done with the desire of destroying the Karm bondage and attaining Moksh with the feeling of surrendering to Hari, that kind of Bhakti is called Saatwik. That is why one should worship Vishnu with Lakshmee."
[From Padm Puraan, 5-Uttar Khand, p. 784]

Navadhaa Bhakti
Bhakti means devotion especially to one's personal Bhagvaan (Isht Dev). A devotee believes that enlightenment follows Bhakti, but not the other way round. Bhakti consists of love and trust. It is not a new phenomenon in Hindu religion, it is as old as Ved. There is a Shlok about nine types of Bhakti -
Shravanam Keertanam Vishnoh Smaranam Paadasevanam
Archanam Vandanam Daasyam Sakhyam Aatmanivedanam.

(1) Shravanam - hearing (hearing good discourses, Bhajan or being near to another person who is singing aloud God's name)
(2) Keertanam - Singing / praising (silently or better still aloud) Devotional songs, Bhajans or divine names...when done aloud, the same person also hears and others who hear also get benefit.
(3) (Vishnoh) Smaranam - Remembering Divine Names or remembering God. Vishnoh refers to Vishnu - meaning All Pervading. God is All Pervading...
(4) Paad Sevanam - worshiping the feet of the Lord and or His Bhakt and or Guru (who is one with God). Try to see the God in every one including the sick, weak, meek and the saints - serve them - Maanav Seva is Maadhav Sevaa and serving poor helpless animals is equal or even higher than helping fellow humans...
(5) Archanam - formal worship to God's photos, idols, Shiv Ling, Shaaligraam, Tulasee plant, Gau Maataa, Atithi, Guru, Saadhus, Sant Mahaatmaa etc. God is formless but some forms of the formless God are specified as more suitable for worship esp. in the early stages of spiritual Saadhanaa. In any case, while worshiping think of God seated / located in the form before you.
(6) Vandanam - bowing with reverence - Doing a paad Namaskaar is also same as Vandanam - one can bow even with one's mind or bow a Vandanam with folded hands....
(7) Daasyam - serving - serving with real love and humility, with full devotion that one is serving the Lord in whatever one is doing...
(8) Sakhyam - friendship - feeling that God is your friend (Arjun is a good example).
(9) Aatm Nivedanam - Total Self Surrender - the highest form of bhakti... accepting every thing as Divine Will and as God's Prasaad...

Nine types of Bhakti told by Shree Raam to Shabaree in Raamaayan, Aranya Kaand.
See here
(1) Satsang (company of good people).
(2) Interest in hearing my Kathaa.
(3) Serving Guru's feet without any pride.
(4) To sing my qualities without cheating.
(5) To do Jaap of my Mantra and strong faith in me - This is famous in Ved.
(6) Control on Indriyaan, good nature and character, Vairaagya from Karm and following saints' behavior.
(7) To see me in the whole world and regard saints more than me. 
(8) To be contented whatever is available and not seeing others faults even in dream.
(9) Plainness and behaving well with everybody without cheating, having trust in me and feeling neither happiness nor sadness in any situation.

Prahlaad describes nine types of Bhakti to Hiranyakashyap -
(1) Shravan - Hearing Bhagvaan's attributes, Leelaa and names - by Hanumaan who always loves to hear Raam Naam
(2) Keertan  - Singing Bhagavaan's attributes, Leelaa and names - This is the simplest form of Bhakti. It was observed by Vaalmeeki, Chaitanya Mahaaprabhu etc Bhakt.
(3) Smaran - Remembering His forms, names etc - Seetaa always remembered Raam's name
(4) Paad Sevya - Serving His feet - Bharat who sacrificed everything in the feet of Raam
(5) Poojaa Archaa - Worshipping Him by offering flowers etc - Gajaraaj did it
(6) Vandan - Worshipping by prayer - Vibheeshan did it for Raam
(7) Daasya - Serving Him with the feeling of slavery - Lakshman did it for Raam
(8) Sakhya - Knowing Him as one's friend - Guh Nishaad who was Raam's friend. Arjun was also Krishn's friend.
(9) Aatm Nivedan - Surrendering to Lord - Jataayu gave even his life to save Raam's wife
[From Bhaagvat Puraan, 7/3]

There is a Shlok also describing these nine types of Bhakti
Shravanam Keertanam Vishnomahaa Smaranam Paad Sevanam
Archanam Vandanam Daasyam Saakhyam Aatm Nivadanam

The first four parts of devotion, like hymns, prayers, remembrance etc are meant for common devotees while the later five parts are meant for higher devotees.

Lord Chaitanya Mahaaprabhu gave 9 steps of Bhakti from beginner to Paramahans......
(1) Shraddhaa
(2) Saadhu-sang
(3) Bhajan-kriyaa
(4) Anarth-nirvritti
(5) Nishthaa
(6) Ruchi
(7) Aasakti
(8) Bhaav
(9) Prem

Study Geetaa, the Raamaayan and the Bhaagavat. Have Satsang. Visit holy places (Teerth Yaatraa). Do Jap. Meditate. Sing His Name. You can develop Bhakti and have His Darshan.

At another place Shree Chaitanya Mahaaprabhu has recommended five most potent forms of Bhakti for everyone, namely (1) Saadhu-Sang -- association with devotees, (2) Naam-Keertan -- chanting the holy names, (3) Bhaagvat-Shravan -- hearing the Bhaagvat, (4) Mathuraa-Vaas -- living in the land of Vraj, and (5) Shree-Moorti-Sevaa -- serving the form of the Lord.

Some Other Types of Bhakti
(1) Nir-Gun Bhakti - Bhakti which is done towards Nir-Gun Bhagavaan
(2) Sa-Gun Bhakti - Bhakti which is done towards a Sa-Gun Roop of Bhagavaan, such as Raam, Krishn, Devee
(3) Ananya Bhakti - An + anya. An means "no", an Anya means "other"; so Ananya means which has no comparison for it.

Difference Between Bhakti and Tantra
Bhakti Maarg is Nish-kaam Bhaav. No desire except to please and do Sevaa of the Lord.
Tanatra is done with Sa-kaam Bhaav - desire to obtain something.

In Bhakti the Lord himself protects the devotee.
In Tantra if you want to protect yourself, you have to Dhaaran Kavach. The strength of the Kavach is the ability and Sadhanaa of Saadhak.